Modern Mechanics and Mathematics – An International Conference in Honour of Ray Ogden’s 60th...

Modern Mechanics and Mathematics

– An International Conference in Honour of

Ray Ogden’s 60th Birthday

Keele University, 26-28 August 2003

ABSTRACTS

1 Combined axial shearing and straightening of elastic an-

nular cylindrical sectors

M. Aron, School of Mathematics and Statistics, University of Plymouth, Plymouth PL4

8AA.

Email: [email protected]

The axial shear deformation of compressible nonlinearly elastic circular cylinders has re-

ceived considerable attention over the past decade. In particular, it was shown by Beatty

& Jiang (1999) and Kirkinis & Ogden (2003) that in an isotropic material this deforma-

tion may coexist with the circular shear deformation of such cylinders, and by Polignone

& Horgan (1992) that this deformation is not possible ( with zero body forces ) in any

Hadamard-Green solid that is not of Neo-Hookean type.

Here we consider the combined axial shearing and straightening of an annular cylindrical

sector which is a deformation that, following Truesedll & Noll (1965) and Hill (1973), we

describe in terms of two prescribed constants and two unknown functions that depend

only on the radial material co-ordinate. Under the assumption that the material is elastic,

compressible and isotropic, we show that for equilibrium in the absence of body forces

the unknown functions must satisfy a system of 1st order non-linear ordinary differential

equations. The system of differential equations can be de-coupled for certain material

classes one of which is the (whole ) class of Hadamard-Green materials. Thus, several

new exact solutions are obtained and, under the assumption that the annular cylindrical

sector is composed of a Hadamard-Green material that is strongly-elliptic, the existence

and uniqueness of solutions for two types of boundary conditions is established.

References

1. Beatty, M.F. and Jiang, Q.: On compressible materials capable of sustaining axisym-

metric shear deformations III. Helical shear of isotropic hyperelastic materials. Quart.

Appl. Math. 57 (1999), 681-697.

2. Hill, J,M.: Partial solutions of finite elasticity- Three dimensional deformations.

ZAMP 24 (1973), 609-618.

1

3. Kirkinis,E. and Ogden, R.W.: On helical shear of a compressible elastic circular

cylindrical tube. Q. Jl. Mech. Appl. Math. 56 (2003), 105-122.

4. Polignone, D.A. and Horgan, C.O.: Axisymmetric finite anti-plane shear of compress-

ible nonlinearly elastic circular tubes. Quart. Appl. Math. 50(1992), 323-341.

5. Truedell, C. and Noll, W.: The non-linear field theories of mechanics. Handbuch der

Physik III/3, ed., S.Flugge, Springer-Verlag, Berlin, 1965.

2 Investigation of mechanical properties of cell membranes

Eveline Baesu, Department of Engineering Mechanics University of Nebraska-Lincoln,

Lexington, KY 40591-0215.


It has been observed that subtle changes of mechanical properties of cells are correlated

with changes in the state of their health. A theory is presented to describe the nonlinear

mechanical properties of living cell membranes, and in particular the response to probing

by Atomic Force Microscopy (AFM). The general theory of liquid crystal bilayer surfaces

with local bending resistance is used in a variational setting to obtain the equations that

describe equilibrium states. This analysis will guide the development of a new generation of

cantilever-based MEMS/NEMS for in vivo/vitro investigation of microbiological systems.

Refinements associated with global constraints on the enclosed volume, and contact with

a rigid substrate, taking the cytoskeleton into consideration are introduced and discussed.

A procedure is also given for identifying material constants for the cell membrane through

correlation with AFM data.

3 Dead loading of a unit cube of compressible isotropic elas-

tic material

R.S. Rivlin and M.F. Beatty, Lehigh University, Bethlehem, PA 18015 and University

of Nebraska-Lincoln, P.O. Box 910215, Lexington, KY 40591-0215.


A unit cube of compressible isotropic elastic material undergoes homogeneous dilatation

by dead loading forces applied to its faces. Conditions are obtained for stability of the

resulting equilibrium state. The physical nature of these conditions is described and the

results are illustrated for a compressible Blatz-Ko foamed rubber material.

4 On Jaeger shear and shearing

Ph. Boulanger and M. Hayes, Departement de Mathematique, Universite Libre de Brux-

elles, Campus Plaine C.P.218/1, 1050 Bruxelles - Belgium, and Department of Mechanical

Engineering, University College Dublin, Belfield, Dublin 4 - Ireland. [email protected]

Email: [email protected], and [email protected]

2

At any point P in a body which is subjected to a finite deformation, the angle between a

pair of material line elements at P is generally changed. The change in angle is called the

“Cauchy shear” of this pair of material line elements. Jaeger introduced another concept of

shear. He considered a material line element and the planar material element orthogonal to

this line element, so that the normal to the planar element is along the line element. After

deformation, the line element and the normal, which were initially along the same direction

make a certain angle. We call this angle the “Jaeger shear” associated with this direction.

Analogously to the definition of Jaeger shear we introduce and examine the concept of

“Jaeger shearing”. It depends upon just one direction, whereas shearing in the sense of

Cauchy depends upon two directions. Results are presented relating the Jaeger shear and

Jaeger shearing to corresponding orthogonal shear and shearing, in the sense of Cauchy, of

appropriate pairs of material line elements. Also it is seen that the maximum Jaeger shear

or Jeager shearing at P at time t is also the maximum Cauchy shear or Cauchy shearing

at P at time t.

5 On maximum shear

Ph. Boulanger and M. Hayes, Departement de Mathematique, Universite Libre de

Bruxelles, Campus Plaine C.P.218/1, 1050 Bruxelles - Belgium, and Department of Mechan-

ical Engineering, University College Dublin, Belfield, Dublin 4 - Ireland. [email protected]


The problem of the determination at any point P in a body of that pair of infinitesimal

material line elements which suffers the maximum shear in a deformation has been solved

[1]. For arbitrary pairs of material line elements, whether orthogonal or not, it was shown

analytically that the pair suffering the greatest shear lies in the principal plane of largest

and least stretch, denoted by λ3 and λ1 respectively, and is symmetrically disposed about

the principal axis corresponding to the least stretch λ1. It subtends the angle Θmax, given

by tan(Θmax/2) = (λ1/λ3)1/2. Also, the maximum shear, denoted by γmax, is γmax =

π − 2Θmax.

Here that problem is revisited and a short proof, of geometrical type, of the result is

presented.

References

1. Ph. Boulanger and M. Hayes, On Finite Shear, Arch. Rational Mech. Anal. 151

(2000), 125–185.

6 Swelling of particle-enhanced elastomers and gels

S. Therkelsen and M.C. Boyce, Department of Mechanical Engineering, Massachusetts

Institute of Technology Cambridge, MA, USA


3

The mechanics of swelling of elastomeric materials has been extensively addressed in the

literature and is reasonably well understood. Recent interest in the mechanics of active

polymers, gels, and soft biological tissues has led to a renewed interest in the mechanics

of swelling of polymeric and polymeric-like materials where reversible swelling is a primary

functional mechanism of many of these materials. Additionally, the properties of elastomers,

active polymers, and gels are often enhanced and selectively tailored by the addition of

particles which act to alter both mechanical and swelling behavior. In this paper, we

study the finite deformation mechanics of swelling of particle-enhanced elastomeric and

elastomeric-like materials. A simple closed form solution for the swelling behavior of the

filled elastomers is presented.

7 On instabilities in pure bending

C. Coman, Department of Mathematics, University of Leicester, Leicester LE1 7RH.


Structural instability is one of the typical failure modes of thin-walled structures. The

importance of this type of failure is shown by extensive numerical and experimental studies.

However, far less attention has been paid to cylindrical shells under the action of pure

bending or transverse shear. In consequence, these problems are poorly understood, at

least from an analytical point of view. In this work we consider a circular cylindrical shell

subjected to a combined loading (bending and transverse shear) and perform an asymptotic

analysis which captures the physics of the problem remarklbly well; numerical results which

back up the analytical study are included as well.

8 Twisting of chiral shafts

M. Fraldi and S. C. Cowin, Dipartimento di Scienza delle Costruzioni, Facolt di Ingeg-

neria, Universit di Napoli “Federico II ”, Italy, and New York Center for Biomedical En-

gineering Departments of Biomedical and Mechanical Engineering The City College 138th

Street and Convent Avenue New York, NY 10031-9198, USA.

Email: [email protected], Web: www.ccny.cuny.edu/NYCBE

Solutions are presented for a class of torsion problems for cylinders of a material with

trigonal material symmetry. In particular the solutions for elliptical, circular and equilateral

triangular cross-sections are presented. These solutions show that the stress distributions

are non-chiral and the same as they would be if the material were isotropic; however the

in-plane displacements are chiral and different from the isotropic case. The results show

that there will be transverse, in-plane stress interactions between the layers of a composite

cylinder composed of concentric cylinders of different trigonal materials in torsional loading.

Such composite cylinders are structural designs used by nature and by man.

References

1. P. Chadwick, M. Vianello and S. C. Cowin, A new proof that the number of linear

anisotropic elastic symmetries is eight, J. Mech. Phys. Solids 49 (2001), 2471-2492.

4

2. S. C. Cowin and M. M. Mehrabadi, On the Identification of Material Symmetry for

Anisotropic Elastic Materials, Quart. J. Mech. Appl. Math., 40 (1987), 451-476.

3. S. C. Cowin and M. M. Mehrabadi, Anisotropic symmetries of linear elasticity, Appl.

Mech. Rev. 48 (1995), 247-285.

4. S. C. Cowin, Elastic symmetry restrictions from structural gradients in, Rational

Continua, Classical and New- A collection of papers dedicated to Gianfranco Capriz

on the occasion of his 75th birthday, (P. Podio-Guidugli, M. Brocato eds.), Springer

Verlag, ISBN 88-470-0157-9, (2002).

5. M. Fraldi and S. C. Cowin, Chirality in the torsion of cylinders with trigonal symme-

try, accepted by Journal of Elasticity.

6. D’arcy Thompson, W. On Growth and Form, Cambridge University Press, Cambridge

(1942).

7. W. Thompson (Lord Kelvin), Baltimore Lectures on Molecular Dynamics and the

Wave Theory of Light, London (1904).

9 On constructing the unique solution for a phase transition

problem: necking in a hyperelastic rod

Hui-Hui Dai and Qinsheng Bi, Department of Mathematics, City University of Hong

Kong,83 Tat Chee Ave., Kowloon Tong, Hong Kong, China.


We use a rod theory to study the probem of the large axially symmetric deformations of a

rod composed of an incompressible Ogden’s hyperelastic material subject to a tensile force

(or a given displacement) when its two ends are fixed to rigid bodies. The attention is on

the class of energy functions for which the strain-stress curve in the case of the uniaxial

tension has a peak and valley combination (typical characteristics of a phase transition

problem). Phase plane analysis is introduced to study the qualitative behaviour of the

solutions and a few theorems are then presented to show the types of the critical points and

their dependence on the physical parameters. Transition boundaries are given to divide

the physical parametric plane into different regions corresponding to qualitatively different

phase planes. In total, we find five types of qualitatively different phase planes. Then, by

using the boundary conditions, the solutions corresponding to trajectories in different phase

planes are obtained and the associated graphic results are presented. It is found that for

certain physical parameters, bifurcations may take place, which lead to jump phenomena

for the deformation with the change of the external force. Furthermore, in the region

bounded by the bifurcation sets, three types of the deformations are found, in two of which

the azimuthal stretch is almost constant in the middle portion of the rod, while the other

type possesses a critical concavity in the middle of the rod, which represents necking. In all

solutions obtained, there is a rapidly changing zone near each end, showing the existence

of a boundary layer.

5

An important and difficult issue in phase transitions is the nonuniqueness of solutions.

Here, by considering the effects of the end boundary layers (which arise due to the nontriv-

ial boundary conditions imposed), our results show that the domain in which the multiple

solutions arise can be much more reduced and further the number of solutions can be re-

duced from four (found in the literature) to three. Further, by converting the problem into

a displacement-controlled problem, the unique solution is obtained. The engineering strain

and engineering stress curve plotted from our solution exhibits the two well-known phenom-

ena observed in experiments: (i)After the stress reaches the peak value there is a sudden

stress drop; (ii)Afterwards it is followed by a stress plateau. Mathematical explanations for

these two phenomena are then given from our model.

10 Interface waves for misaligned deformed incompressible

half-spaces

Michel Destrade, Laboratoire de Modelisation en Mecanique, CNRS, UMR 7607, Uni-

versite Pierre et Marie Curie, 4 Place Jussieu, Tour 66, Case 162, 75252 Paris Cedex 05,

France

Email: [email protected], Web: www.lmm.jussieu.fr/ destrade

Some relationships, fundamental to the resolution of interface wave problems, are pre-

sented. These equations allow for the derivation of an explicit secular equation and explicit

displacement components for problems involving waves localized near the plane bound-

ary of anisotropic elastic or viscoelastic half-spaces, such as Rayleigh, Sholte, or Stoneley

waves. They are obtained rapidly, without using the Stroh formalism. As an application,

the problems of Stoneley wave propagation and of interface stability for misaligned prede-

formed incompressible half-spaces are treated. The upper and lower half-spaces are made

of the same material, subject to the same prestress, and are rigidly bonded along a common

principal plane. The principal axes in this plane do not however coincide, and the wave

propagation is studied in the direction of the bisectrix of the angle between a principal axis

of the upper half-space and a principal axis of the lower half-space.

11 Null condition for nonlinear elastic materials

Wlodzimierz Domanski and Ray Ogden, Institute of Fundamental Technological Re-

search, Polish Academy of Sciences, ul. Swietokrzyska 21, 00-049 Warsaw, Poland, and

Department of Mathematics, University of Glasgow, Glasgow G12 8QW

Email: [email protected] and [email protected]

Smooth solutions to the Cauchy problem for the equations of nonlinear elastodynamics

exist typically only locally in time. However, under the assumption of small initial data

and an additional restriction, the so-called null condition, global existence of a classical

solution can be proved.

We investigate this condition and its connection with the property of genuine nonlin-

earity. We also discuss its connection with the phenomenon of nonlinear wave resonance.

Moreover, we analyse the null condition for different type of elastic materials, including

6

some models for soft tissues and rubberlike materials. This allows us to formulate crite-

ria for existence of classical solutions to the initial value problem for the elastodynamics

equations in terms of the strain energy for these nonlinear models.

12 The Pseudo-elastic response of rubberlike solids

Luis Dorfmann, Institute of Structural Engineering, Peter Jordan Street 82, 1190 Vienna.


The seminar focuses on the mechanical behaviour and on important aspects of material

modelling of filled and unfilled natural rubber. This interest has been generated by the in-

creasing use of elastomers, for example in vibration isolators, vehicle tires, shock absorbers,

earthquake bearings and others. Filled and unfilled elastomers under cyclic loading show no-

ticeable differences between the mechanical response under loading and unloading during

the first cycles in oscillation tests. We examine the change in material response associ-

ated with the Mullins effect and with cavitation nucleation arising from tensile hydrostatic

stresses of sufficient magnitude. The second part of this seminar focuses on the formulation

of constitutive equations using the theory of pseudo-elasticity due to Ogden and Roxburgh

(1999). The basis of this theory is the inclusion of a damage variable in the strain-energy

function W. Specifically, the strain-energy function of an elastic material depends on a

scalar parameter, which provides a means for modifying the form of the strain-energy func-

tion, thereby reflecting the stress softening associated with unloading and the accumulation

of residual strains. The dissipation of energy, i.e. the difference between the energy input

during loading and the energy returned on unloading is also accounted for in the model

by the use of a dissipation function, which evolves with the deformation history. A good

correspondence between the theory and the data is obtained.

References

1. Ogden, R.W., Roxburgh, D.G., A pseudo-elastic model for the Mullins effect in filled

rubber. Proc. R. Soc. Lond. A 455 (1999), 2861-2878.

13 On a class of inhomogeneous deformations controllable in

isotropic incompressible elastic solids

J. Dunwoody, School of Mathematics and Physics, The Queen’s University of Belfast,

Belfast BT7 1NN, UK.


In §59 of Truesdell & Noll (1965) a class of deformations involving one or more unknown

functions was proposed for consideration as statically possible in isotropic, incompressible

elastic materials. One or more of the unknown functions must be determined using semi-

inverse methods by solution of non-linear ordinary differential equations arising from the

equations for static equilibrium for specific materials. Saccomandi (1996) examined this

7

class of deformations in perfectly elastic inhomogeneous materials, the inhomogeneity aris-

ing from ‘layering’ due to a temperature gradient. He restricted his analysis to physical

problems involving rectangular Cartesian coordinates. By adopting Fourier’s law of heat

conduction in which the heat conductivity is a scalar constant, Saccomandi (1996) was able

to determine the temperature and hence the nature of the inhomogeneity independently

of the deformation. He then solved specific boundary value problems for neo-Hookean

materials

Here we consider this class of deformations without the restriction to rectangular Carte-

sian coordinates. Two forms of Fourier’s law of heat conduction which allow the determi-

nation of the temperature to be independent of the deformation are proposed. A form of

inherent inhomogeneity due to the material having a layered structure, the layers being in-

finitesimally thin, is also considered (cf. Wang 1968 and Bilgili et. al. 2003). In the presence

of either type of inhomogeneity, it is deduced using the criteria of Dunwoody (2003) that

none of the deformations belong to any of the families of universal deformations which are

controllabe in homogeneous materials. Three subclasses of the general class of deformations

are defined. Within these subclasses, existence of controllable solutions to the equations of

static equilibrium involving the unknown functions is established for neo-Hookean materials

using conventional inequalities, and more generally for materials satisfying the constraints

of the well known Baker-Ericksen inequalities.

Acknowledgement: This work has been supported by a grant, GR/27196, from the EPSRC,

UK.

References

1. C. Truesdell and W. Noll.: Non-linear field theories of mechanics. Handbuch der

Physik, III/3, ed. S. Flugge. Springer-Verlag. Berlin-Heidelberg-New York, 1965.

2. G. Saccomandi.: A note on inhomogeneous deformations of nonlinear elastic layers.

IMA Journal of Applied Mathematics 57 (1996), 311–324.

3. C. C. Wang.: Universal solutions for incompressible laminated bodies. Archive for

Rational Mechanics and Analysis 29 (1968), 161–173.

4. E. Bilgili, B. Berstein and H. Arastoopour.: Effect of material inhomogeneity on

the inhomogeneous shearing deformation of a Gent slab subjected to a temperature

gradient. International Journal of Non-linear Mechanics 38 (2003), 1351–1368.

5. J. Dunwoody.: On universal deformations with non-uniform temperatures in isotropic,

incompressible elastic solids. Mathematics and Mechanics of Solids 8 (2003), in press.

6. M. Baker and J. L. Ericksen.: Inequalities restricting the form of the stress defor-

mation relations for isotropic elastic solids and Reiner-Rivlin fluids. Journal of the

Washington Academy of Science 44 (1954), 33–35.

8

14 Dynamic extension of a compressible nonlinearly elastic

membrane tube

H.A. Erbay and V. H. Tuzel, Department of Mathematics, Faculty of Science and

Letters, Istanbul Technical University, Maslak 34469, Istanbul, Turkey, and Department

of Mathematics, Faculty of Science and Letters, Isık University, Maslak 80670, Istanbul,

Turkey.


The dynamic response of an isotropic compressible hyperelastic membrane tube, subjected

to a dynamic extension at its one end, is studied. The analysis contained in the present study

parallels quite closely that described in Tuzel and Erbay (2003) where the same problem

has been studied for an incompressible hyperelastic membrane tube. The main difference

between Tuzel and Erbay (2003) and the present study arises in consideration of the tube

material. Here we consider a circular cylindrical tube composed of a general compressible

hyperelastic material. For incompressible hyperelastic membrane tubes, problems of this

type were first discussed by Tait and Zhong (1994a, b).

In the first part of the study, an asymptotic expansion technique is used to derive a

nonlinear membrane theory for finite axially symmetric dynamic deformations of compress-

ible nonlinearly elastic circular cylindrical tubes by starting from the three-dimensional

elasticity theory. The equations governing dynamic axially symmetric deformations of the

membrane tube are obtained for an arbitrary form of the strain-energy function. In the

second part of the study, finite amplitude wave propagation in a compressible hyperelastic

membrane tube is considered when one end is fixed and the other is subjected to a suddenly

applied dynamic extension. The equations of motion along with compatibility conditions

are written as a quasilinear hyperbolic system of first-order partial differential equations.

A Godunov-type finite volume method is used to solve numerically the corresponding prob-

lem. Numerical results are given for both the neo-Hookean compressible material and the

Blatz-Ko compressible material. The question how the present numerical results are related

to those obtained for incompressible materials in the literature is discussed.

References

1. V. H. Tuzel and H. A. Erbay, The dynamic response of an incompressible non-linearly

elastic membrane tube subjected to a dynamic extension. Int. J. Non-Linear Mech.

(in press).

2. R. J. Tait and J. L. Zhong, Wave propagation in a non-linear elastic tube. Bull. Tech.

Univ. 47 (1994a), 127.

3. R. J. Tait and J. L. Zhong, Dynamic extension and twist of a non-linear elastic tube.

Int. J. Non-Linear Mech. 30 (1994b), 887.

9

15 On travelling wave solutions of a generalized Davey-Stewartson

system

Alp Eden and Saadet Erbay, Department of Mathematics, Bogazici University, Istanbul,

Turkey, and Department of Mathematics, Isik University, Istanbul, Turkey.


In a recent study [1], coupled evolution equations that may be called generalized Davey-

Stewartson (GDS) equations were derived

iuτ + uξξ + γuηη = χ|u|2u + b(ϕ1,ξ + ϕ2,η)u,

ϕ1,ξξ + m2ϕ1,ηη + nϕ2,ξη = (|u|2)ξ,

λϕ2,ξξ + m1ϕ2,ηη + nϕ1,ξη = (|u|2)η. (1)

The system (1) involves three equations, two for the long waves, ϕ1 and ϕ2, and one for

the short wave, u, propagating in an infinite homogeneous elastic medium. The GDS

system was classified in [2] according to the values of its parameters as hyperbolic-elliptic-

elliptic, hyperbolic-hyperbolic-hyperbolic and hyperbolic-elliptic-hyperbolic. Special trav-

elling wave solutions to GDS were exhibited in [1] that were of sech-tanh-tanh and tanh-

tanh-tanh forms. In this note, we first seek the validity of solutions within the classification

scheme, then establish via Pohazaev-type identity the non-existence of travelling waves in

the elliptic-elliptic-elliptic case. In a similar manner, in the hyperbolic-elliptic-hyperbolic

case some specific parameter constraints are introduced as necessary conditions for the

existence of travelling waves.

References

1. C. Babaoglu and S. Erbay, Two-dimensional wave packets in an elastic solid with

couple stresses, Int. J. Non-Linear Mechanics, (in press, 2003).

2. C. Babaoglu, A. Eden and S. Erbay, A blow-up result for a generalized Davey-

Stewartson system, (submitted, 2003).

16 On phase transitions in nonlinear elastic media and struc-

tures

Victor A. Eremeyev, Mechanics and Mathematics Department of Rostov State Univer-

sity, Zorge str., 5, Rostov-on-Don, 344090, Russia.


For stress-induced phase transformations in solids, the mathematical model is proposed

on the base of Gibbs’s variational principles. The constitutive equations of the 2D and

3D micropolar elastic media under finite deformations are considered. Each point of the

micropolar media has additional rotational degrees of freedom. This model possesses couple

stresses and takes into consideration orientational interaction of material particles. The

mathematical models based on the theories of polar media have significant applications to

10

description of real materials with microstructure such as composites, granular materials,

nanostructures, magnetic fluids, liquid crystals.

The equilibrium conditions of two-phase body are obtained. These conditions consist

of equilibrium equations in phase volumes and the boundary relations at the phase surface.

The last relations describe the balance of forces and couples and contain the relation that

is required to determine the a priory unknown phase surface. For the micropolar media

the energy-momentum tensors are introduced. As an example the phase transformation in

bodies with dislocations is investigated.

Within the framework of the general non-linear theory of shells (2D micropolar con-

tinuum), thermodynamical equilibrium conditions are derived for a shell undergoing phase

transition of martensitic type. Following the variational methods, the balance equations at

the phase separation curve are obtained. For elastic micropolar shells, the energy-impulse

tensor is introduced. Some applications to modelling of thin films made of shape-memory

alloys like NiTi are considered. Such thin films are among the best for production of

micro-actuators in micro-electro-mechanical systems.

The proposed models may be useful to description of the phase and structural transitions

of orientational type in solids and thin-walled structures.

17 Equilibrium spherically-symmetric two-phase deforma-

tions of nonlinear elastic solids within the frameworks of

phase transition zone

A. B. Freidin and Y.B. Fu, Institute of Mechanical Engineering Problems, Russian Academy

of Sciences, Bolshoi pr. 61, V.O., St. Petersburg 199178, Russia, and Department of Math-

ematics, Keele University, Staffordshire ST5 5BG, UK.

Email: [email protected], and [email protected].

We study two-phase spherically symmetric deformations that can be supported by a non-

linear elastic isotropic material. We develop a general procedure for the construction of the

solution for an arbitrary nonlinear elastic material. Then we study stress-induced phase

transformations for the Hadamard material. We demonstrate that even in this simplest

case the solution is not unique. Two different equilibrium two-phase states as well as a

uniform one-phase state can be found under the same boundary conditions. We show that

one of the two-phase solutions is unstable. The stability properties of the other two-phase

solution are not yet entirely clarified, but it is shown that the energy of this solution is

less than the energy of the one-phase solution. We study characteristic features of the

distribution of deformations in an equilibrium two-phase body in detail. Then we consider

the spherically symmetric solutions in the context of a phase transition zone (PTZ) formed

in a strain space by all deformations which can exist on the equilibrium phase boundary

(Freidin and Chiskis 1994, Freidin et al. 2002). The PTZ boundary acts as a phase diagram

or yield surface in strain-space. We study how deformations associated with each solution

are related with the PTZ. Finally, we compare our results with the results obtained earlier

by a small strain approach (Morozov et al. 1996, Nazyrov and Freidin 1998).

The work is supported by the Royal Society and the Russian Foundation for Basic

11

Research (Grant N 01-01-00324).

References

1. A.B. Freidin and A.M. Chiskis Regions of phase transitions in nonlinear-elastic isotropic

materials. Part 1: Basic relations. Izvestia RAN, Mekhanika Tverdogo Tela (Mechan-

ics of Solids), Vol. 29, No. 4 (1994), 91-109. Part 2: Incompressible materials with

a potential depending on one of deformation invariants. Izvestia RAN, Mekhanika

Tverdogo Tela (Mechanics of Solids) Vol. 29 (1994) No. 5, 46-58.

2. A.B. Freidin, E.N. Vilchevskaya and L.L. Sharipova. Two-phase deformations within

the framework of phase transition zones. Theoritical and Apllied Mechanics, Vol.

28-29 (2002), 149-172.

3. N.F. Morozov, I.R. Nazyrov and A.B. Freidin, One-dimensional problem on phase

transformation of an elastic sphere. Doklady Akademii Nauk (Proceedings of the

Russian Academy of Sciences). Vol. 346 (1996), No. 2, 188-191.

4. I.R. Nazyrov and A.B. Freidin, Phase transformation of deformable solids in a model

problem on an elastic sphere. Izvestia RAN, Mekhanika Tverdogo Tela (Mechanics

of Solids). Vol. 33 (1998), No. 5, 52-71

18 On the stability of piecewise-homogeneous deformations

Y.B. Fu and A. B. Freidin, Department of Mathematics, Keele University, Staffordshire

ST5 5BG, UK, and Institute of Mechanical Engineering Problems, Russian Academy of

Sciences, Bolshoi pr. 61, V.O., St. Petersburg 199178, Russia.

Emails: [email protected], and [email protected]

Many solid materials exhibit stress-induced phase transformations. Such phenomena can

be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the

strain-energy function. It was shown by Gurtin (1983) that if a two-phase deformation

(with gradient F) in a finite elastic body is a local energy minimizer, then given any point

p0 of the surface of discontinuity, the piecewise-homogeneous deformation corresponding to

the two values F±(p0) of F(p0) is a global energy minimizer. Thus, instability of the latter

state would imply instability of the former state. In this paper we investigate the stability

and bifurcation properties of such piecewise-homogeneous deformations. More precisely, we

are concerned with two joined half-spaces that correspond to two different phases of the

same material. We first show how such a two-phase deformation can be constructed. Then

we determine the condition under which such a two-phase piecewise-homogeneous deforma-

tion bifurcates into an inhomogeneous deformation with a wavy interface, the incremental

inhomogeneous deformation decaying to zero exponentially away from the interface. The

stability of the piecewise-homogeneous deformation is investigated with the aid of two cri-

teria. One is a dynamic stability criterion based on a quasi-static approach; the other is

by determining whether the deformation is a minimizer of the potential energy. The two

criteria are found not to coincide with each other. A numerical example is used to show

that when perturbations/variations of the interface in the undeformed configuration are

12

allowed, the region of stability (when either stability criterion is used) is only a subset of

the corresponding region of stability when such perturbations/variations of the interface

are not allowed.

References

1. V.A. Eremeyev, On the stability of nonlinear elastic bodies with phase transfor-

mations. Proc. 1st Canadian conference on nonlinear solids mechanics (ed. E.M.

Croitoro), Vol.2 (1999), 519-528.

2. V.A. Eremeyev and L.M. Zubov, On the stability of equilibrium of nonlinear elastic

bodies with phase transformations. Proc. USSR Academy of Science. Mech. Solids

(1991), 56-65 (in Russian).

3. V.A. Eremeyev, A. Freidin and L. Sharipova, On nonuniqueness and stability of cen-

trally symmetric two-phase deformations. Proc. “Advanced Problems in Mechanics”

Conference 2001 (eds V.A. Palmov and D.A. Indeitsev), 2001, 198-206.

4. A.B. Freidin and A.M. Chiskis, Phase transition zones in nonlinear elastic isotropic

materials. Part 1: Basic relations. Izv. RAN, Mekhanika Tverdogo Tela (Mechanics

of Solids) 29 (1994), 91–109.


materials. Part 2: Incompressible materials with a potential depending on one of

deformation invariants. Izv. RAN, Mekhanika Tverdogo Tela (Mechanics of Solids)

29 (1994), 46–58.

6. Y.B. Fu and A. Mielke, A new identity for the surface-impedance matrix and its

application to the determination of surface-wave speeds. Proc. Roy. Soc. Lond.

A458 (2002), 2523-2543.

7. M.E. Gurtin, Two–phase deformations of elastic solids. Arch. Rat. Mech. Anal. 84

(1983), 1–29.

19 Vibrations of layered thermoelastic continua

M. Gei, D. Bigoni, and G. Franceschini, Department of Mechanical and Structural

Engineering, University of Trento Via Mesiano, 77, I-38050 Trento, Italy.

[email protected], Web: www.ing.unitn.it/ mgei

A framework for thermoelastic analysis of wave propagation in multilaminated struc-

tures is given. Layered, compressible, nonlinear materials, described by a free-energy func-

tion within the modified entropic theory, are considered, deformed an arbitrary amount

with deformations having principal Eulerian axes aligned parallel and orthogonal to the

layers. Temperature is assumed uniform in this configuration and equal in all layers. Ther-

moelastic, plane strain, and small-amplitude waves are analyzed from this state, in a fully

coupled formulation. Within the analyzed range of parameters, it is shown that the cou-

pling terms, yielding complex propagation velocities, introduce a small dispersion effect.

13

However, temperature and pre-strain result to play an important role in determining the

propagation characteristics of the structures.

20 Simple models for rebound

R. J. Knops and Piero Villaggio, Department of Mathematics, Heriot-Watt University,

Edinburgh EH14 4AS, and Dipartimento di Ingegneria Strutturale, Universita degli Studi

di Pisa, via Diotisalvi, 2 56126 Pisa.

Three simple models are discussed to help explain the process that occurs when de-

formable bodies rebound on impact. The models assume the colliding bodies are a one-

dimensionalised rod and half-space, and the rod to be (a)linear elastic (b)elastic-plastic

(c)rigid. The half-space is rigid for (a) and (b), while it is supposed elastic for (c). Appro-

priate factors, such as the time of rebound, are calculated in each example.

21 Some properties of a new model for slow flow of granular

materials

D. Harris, Department of Mathematics, UMIST, Manchester M60 1QD.


The problem of constructing a continuum model for the bulk flow of a dense granular mate-

rial in which neighbouring grains are in contact for a finite duration of time and in which the

contact force is non-impulsive - the so called slow flow regime - has proven to be both a dif-

ficult and controversial problem. There is no consensus of opinion on many basic issues, for

example, there is no agreement as to whether the governing equations should be well-posed.

Many models exhibit a particular form of linear ill-posedness, for example the plastic poten-

tial model for non-associated flow rules and the double-shearing model, which implies that

solutions are unstable, and that the instability is of a particularly strong form. A model

of slow granular flow with a domain of well-posedness is presented here. The equations

generalise both the plastic potential and double-shearing models and contain an additional

kinematic quantity - the intrinsic spin. The stress tensor is, in general, non-symmetric and

a second yield condition governs the rotational yield. The problem of dilatant simple shear

flow is considered here and it is demonstrated that dilatant/contractant flows are unstable

and that of the simple shear flows for the plastic potential and double- shearing models,

the former is stable and the latter is unstable.

22 On rigid-elastic bending and buckling deformations of

long beams

K.A. Lazopoulos, Mechanics Division, School of Applied Sciences, National Technical

University of Athens, Zografou Campus, Athens, Greece 157 73.


Localized bending and buckling of long beam-like straight films due to the change of stiffness

14

is presented. A two-phase beam model is developed. The one phase is considered of infinite

stiffness (rigid deformation). The localized phase is studied and the rigid deformation is

defined. This kind of two -phase deformations may be exhibited in thin surface structures

such as films.

23 Recovery of residual stress in a vertically heterogeneous

elastic medium

Sergei A. Ivanov, Chi-Sing Man, and Gen Nakamura, Russian Center of Laser

Physics, St. Petersburg University, St. Petersburg, 198904, Russia, Department of Mathe-

matics, University of Kentucky, Lexington, Kentucky 40506-0027, USA, and Department of

Mathematics, Graduate School of Sciences, Hokkaido University, Sapporo, 060-0810, Japan.


We study the problem of identifying residual stress in an elastic medium occupying a region

Ω = (x1, x2, x3) ∈ R3 : 0 < x3 < L,where L ≤ ∞ in space, where all parameters depend

only on the depth x3. Under the theoretical framework of linear elasticity with initial stress,

the incremental elasticity tensor of each material point is written as a sum of two terms,

namely the elasticity tensor and the acoustoelastic tensor, both of which are taken here as

isotropic functions of their arguments. Giving some loads and measuring the displacements

at the boundary, we recover the residual stress and its gradient at the boundary x3 = 0. If

the residual stress has a diagonal form, we can recover the residual stress inside the medium.

24 Propagation of waves in composites, high-order homoge-

nization and phononic band gap structures

A.B. Movchan, Department of Mathematical Sciences, University of Liverpool, Liverpool

L69 3BX.


This lecture includes results of the recent work based on analysis of mathematical models

of elasticity describing Bloch waves in doubly periodic structures. The work includes the

following three parts:

1. The background model incorporates the spectral problem for the Navier system posed

in a region containing a doubly periodic array of circular voids or elastic inclusions. The

Bloch-Floquet conditions are set on the boundary of an elementary cell, and the Neumann

boundary conditions are prescribed on the contour of voids (for the case of elastic inclu-

sions, we prescribe transmission conditions that represent continuity of displacements and

tractions across the interface). Pressure and shear waves are coupled via the boundary con-

ditions, and the waves propagating within such a system are dispersive. The eigen-solutions

are represented by multipole series, and an accurate algorithm has been developed for anal-

ysis of the dispersion equation. When the inclusions/voids are sufficiently close to each

other, the stop bands appear in the dispersion diagram which indicates that no waves of

15

given polarisation can propagate through the periodic structure within certain range of fre-

quencies. The work covers both cases of transverse and oblique incidence. 2. The spectral

analysis is complemented by the study of scattering problems for stacks of elastic inclusions.

The analytical model has been developed for evaluation of transmission and reflection co-

efficients characterising the interaction of elastic waves with the stack. 3. The final part of

the talk will include analysis of structures with defects and discussion of the coupling effects

involving electromagnetic and elastic waves. The model enables one to make an accurate

prediction of frequencies corresponding to localised dilatational modes and to explain the

important experimental observations made by P.St.J.Russell and his colleagues.

In addition, I will show how to use simple discrete lattice approximations for analysis

of phononic band gaps in doubly periodic elastic structures.

25 Inhomogeneity, couple-stress, and time-dependent mate-

rial systems – a molecular-based continuum viewpoint

Ian Murdoch, Department of Mathematics, University of Strathclyde, Livingstone Tower,

26 Richmond street, Glasgow G1 1XH.


A procedure for the derivation of continuum equations of balance from a simple model

of molecular behaviour will be outlined. Particular attention will be paid to highly-

inhomogeneous and time-dependent material systems such as are encountered in crack

propagation and at phase interfaces, motivated by the work of Gurtin and Maugin on

so-called ’configurational forces’.

26 Inhomogeneous prestressing of cylindrical tubes

Jerry Murphy, Department of Mechanical Engineering, Dublin City University, Glas-

nevin, Dublin 9, Ireland.


The bending of a cylindrical sector so that it forms a cylindrical tube has been recently

proposed as a method of introducing an inhomogeneous prestress in a tube of an incom-

pressible, homogeneous, isotropic, elastic material. The effect of this prestress on some

qualitative features of the behaviour of such tubes, and, in particular, the response of the

tube to an internal pressure, will be explored. Some non-uniqueness issues will also be

discussed. The behaviour of incompressible materials will be contrasted with that of some

special compressible materials for which solutions describing the bending of cylindrical sec-

tors have also been recently obtained.

27 Swelling induced cavitation of elastic spheres

Thomas J. Pence and Hungyu Tsai, Department of Mechanical Engineering, Michigan

State University, East Lansing, MI 48824, USA.


16

Swelling, generally referring to the volumetric change due to mass addition resulting from

a variety of diffusive and transport mechanisms, is central to a variety of physical phenom-

ena. Here we discuss a mathematical framework for the swelling of elastic solids within the

setting of finite deformation continuum mechanics. The general framework is based on the

minimization of potential energy that prefers the locally prescribed swollen state. We also

consider a material that behaves otherwise incompressibly in that the volume change is

dictated by the given swelling field. The treatment follows that of incompressible, isotropic

hyperelasticity with a local volume constraint representing the additional swelling field.

This framework is then applied to the case of spherical symmetry so as to treat a prob-

lem that has been extensively studied in the classical theory of isotropic, incompressible

hyperelasticity, namely cavity formation at the center of a solid sphere due to radially sym-

metric tensile load on the outer surface. In the extended theory that includes swelling,

both load and swelling can drive the cavitation processes. Further, cavitation can be driven

by swelling alone in the absence of load. Specifically, we consider a two-zone, piecewise

constant swelling field wherein the outer portion of the sphere swells more than the inner

core. The problem is formulated in terms of the inward advance of a spherically symmetric

swelling front separating the outer and inner swelling zones. For sufficiently high swelling

ratio (outer/inner), cavity nucleation is found to occur at the sphere center as the swelling

front advances past a critical radius. The continued advance of the swelling front gives an

initial period of cavity growth followed by a secondary period of cavity collapse with cavity

disappearance as the front approaches the cavity surface.

28 On quasi-fronts in a bi-axially pre-stressed incompressible

plate

A. V. Pichugin, J. D. Kaplunov and G. A. Rogerson, Department of Mathematics,

University of Manchester, M13 9PL, and Mathematics, School of Sciences, University of

Salford, Salford M5 4W.


A refined long-wave low-frequency theory is used to investigate the far field response of a bi-

axially pre-stressed incompressible plate subjected to the instantaneous impulse loading at

an edge point. Whereas the leading order plate theory is hyperbolic and predicts undistorted

propagation of wave fronts, the higher-order derivative terms introduced within the refined

theory produce the boundary layers, which smooth the discontinuities associated with wave

fronts. The described quasi-fronts are studied using the method of matched asymptotic

expansions. The explicit analytic solutions for the vicinity of quasi-fronts are obtained and

analysed numerically. The influence of pre-stress is most strikingly demonstrated by the

presence of bending quasi-front that has no analogue in isotropic theory. It is also possible

to vary pre-stress to modify the type of generated quasi-front from the classical receding to

the advancing.

17

29 A WKB analysis of the buckling of a cylindrical shell of

arbitrary thickness

M. Sanjarani Pour, Mathematics Department, Science College, Sistan & Baluchestan

University, Zahedan, IRAN.


In this paper we apply a full asymptotic analysis to the plane-strain buckling of a cylindrical

shell of arbitrary thickness, which is subjected to an external hydrostatic pressure on its

outer surface. The material of the cylindre is Varga. We follow Fu (1998) and Fu &

Sanjarani Pour (2001) and use the WKB solution in such an equivalent form in order to be

able to solve the eigenvalue problem. Symmetric buckling takes place at a value of µ1 which

depends on A1/A2 and the mode number, where A1 and A2 are the undeformed inner and

outer radii and µ1 is the ratio of the deformed inner radius (a1) to the undeformed inner

radius.

We show that for large mode numbers, the dependence of µ1 on A1/A2 has a boundary-

layer structure. It is independent of the thickness of the shell and is constant over almost

the entire region of 0 < A1/A2 < 1 and decreases sharply from this constant value to

unity as A1/A2 tends to unity. The existence of a second solution can also be confirmed

by the argument of Ogden and Roxburgh (1993) or Rogerson and Fu (1995) that in the

large mode number limit, the small wavelength buckling modes do not feel the curvature

of the cylindrical tube and so the tube is like a flat plate with respect to such modes. It

is known that a pree-stressed plate can suport two types of buckling modes, one is flexural

and the other extensional. The main solution found over the entire region mentioned above

corresponds to the flexural mode whereas the second solution corresponds to the extensional

mode. Asymptotic results for A1−1 = O(1) and A1−1 = O(1/n) agree with the numerical

results obtained by using the compound matrix method.

References

1. Y.B. Fu, Some asymptotic results concerning the buckling of a spherical shell of

arbitrary thickness. Internat. J. Non-Linear Mech. 33 (1998), 1111-1122.

2. Y.B. Fu and M. Sanjarani Pour, WKB method with repeated roots and its application

to the buckling analysis of an everted cylindrical tube. SIAM J. Appl. Math. 62

(2002), 1856-1871.

3. R.W. Ogden and D.G. Roxburgh, The effect of pre-stress on the vibration and stability

of elastic plates. Int. J. Engng Sci. 31 (1993), 1611-1639.

4. G.A. Rogerson and Y.B. Fu, An asymptotic analysis of the dispersion relation of a

pre-stressed incompressible elastic plate. Acta Mechanica 111 (1995), 59-74.

18

30 Low and high frequency motion in compressible finitely

deformed elastic layers

G. A. Rogerson and L. A. Prikazchikova , Department of Computer and Mathemat-

ical Sciences University of Salford Salford M5 4WT UK.


The dispersion of small amplitude waves in a compressible, finitely deformed elastic layer,

with incrementally traction-free upper and lower surfaces, is investigated The associated dis-

persion relation is derived and numerical solutions presented in respect of two-dimensional

motions. The main goal of the work is to derive asymptotically consistent models for low and

high frequency long wave motion. To achieve this, appropriate long wave approximations of

the dispersion relation are first established. These are then used to estimate the orders of the

displacement components. After gaining knowledge of the relative orders of displacements,

and introducing appropriate space and time scales, approximate governing equations are

established. To illustrate the main results attention is focussed on anti-symmetric motion.

In the case of anti-symmetric low frequency motion, asymptotic integration of the appro-

priate approximate equations results in a leading order one-dimensional string-like theory.

A higher order string-like equation, containing fourth order derivatives, is also derived.

In the vicinity of the so-called quasi wave front, this higher order governing equation for

the mid-surface deflection becomes asymptotically leading. A simple edge-loading problem

for a semi-infinite plate is set up and solved to illustrate the theory. In the case of high

frequency motion, asymptotic models are established for motion within the vicinity of the

various families of cut-off frequencies. In contrast to previous studies, for the corresponding

incompressible problem, both thickness stretch and thickness shear resonance are observed

to be possible.

31 Superposition of generalized plane deformations with anti-

plane shear deformations with in isotropic incompressible

hyperelastic materials

G. Saccomandi, Dipartimento di Scienza dei Materiali, Universit di Lecce, Italy.


The purpose of this research is to investigate the basic issues that arise when generalized

plane deformations are superimposed on anti-plane shear deformations in isotropic incom-

pressible hyperelastic materials. Attention is confined to a subclass of such materials for

which the strain-energy density depends only on the first invariant of the strain tensor. The

governing equations of equilibrium are a coupled system of three nonlinear partial differ-

ential equations for three displacement fields. It is shown that this system decouples only

the plane and ant-plane displacements for the case of a neo-Hookean material. Even in this

case, the stress field involves coupling of both deformations. For generalized neo-Hooken

materials, universal relations may be used in some situations to uncouple the governing

equations. It is shown that some of the results are also valid for inhomogeneous materials

and for elastodynamics.

19

32 Wave stability for constrained materials in anisotropic

generalised thermoelasticity

N.H. Scott, School of Mathematics, University of East Anglia, Norwich NR4 7TJ.


In generalised thermoelasticity Fourier’s law of heat conduction in the classical theory

of thermoelasticity is modified by introducing a relaxation time associated with the heat

flux. Equations are derived for the squared wave speeds of plane harmonic body waves

propagating through anisotropic generalised thermoelastic materials subject to thermome-

chanical constraints of an arbitrary nature connecting deformation with either tempera-

ture or entropy. In contrast to the classical case, it is found that all wave speeds remain

finite for large frequencies. As in the classical case, it is found that with temperature-

deformation constraints one unstable and three stable waves propagate in any direction

but with deformation-entropy constraints there are three stable waves and no unstable

ones. Many special cases are discussed including purely thermal and purely mechanical

constraints.

33 Equilibrium two-phase deformations and phase transition

zones in a small strain approach

Leah L. Sharipova and Alexander B. Freidin, Institute of Mechanical Engineering Prob-

lems, Russian Academy of Sciences, Bolshoi pr. 61, V.O., St. Petersburg 199178, Russia


If phase transitions take place in some parts of a deformable body, phase boundaries can

be considered as surfaces across which the deformation gradient suffers a jump and dis-

placements are continuous. On the equilibrium interface a thermodynamic condition has

to be put in addition to conventional displacement and traction continuity conditions. The

thermodynamic condition can be satisfied not by any deformation on the interface. The

deformations which can coexist on the equilibrium phase boundary form the phase transi-

tion zone (PTZ) (Freidin and Chiskis 1994, Morozov and Freidin 1998, and Freidin et al.

2002). The PTZ boundary acts as a yield surface or phase diagram in strain-space.

In this paper a procedure for the PTZ construction is developed by a small-strain

approach. It is demonstrated that different types of strain localization due to phase trans-

formation are possible on different loading paths. Depending on material parameters, the

PTZ can be closed or unclosed. The last means that phase transitions are impossible on

some deformation paths. A model of phase transformation due to multiple appearance of

new phase layers is developed. Paths of transformation are related with the PTZ. Average

stress-strain diagrams on the path of transformation are constructed. Effects of internal

stresses induced by new phase areas and the anisotropy of a new phase are discussed.

This work was supported by Russian Foundation for Basic Research (Grants N 01-01-

00324, 02-01-06263).

20

References


materials. Part 1: Basic relations. Izv. RAN, Mekhanika Tverdogo Tela (Mechanics

of Solids) 29 (1994) No. 4, 91-109.

2. N.F. Morozov and A.B. Freidin, Zones of phase transition zones and phase trans-

formations in elastic bodies under various stress states. Proceedings of the Steklov

Mathematical Institute, 223 (1998) 219–232.

3. A.B. Freidin, E.N. Vilchevskaya, L.L Sharipova, Two-phase deformations within the

framework of phase transition zones. Theoretical and Apllied Mechanics, 28-29

(2002) 149–172.

34 Convergence of regularised minimisers in finite elasticity

J. Sivaloganathan, Department of Mathematical Sciences, University of Bath, Bath BA2

7AY, UK.


Consider a hyperelastic body which occupies the domain Ω in its reference configuration

and which is held in a state of tension under imposed boundary displacements. Let x0 ∈ Ω

be given (this represents one of possibly many flaws in the material). It is shown in [1] that

there exists a minimiser of the energy in a class of deformations containing maps which

may be discontinuous at x0. For weak materials it is known that any such minimiser must

be discontinuous if the imposed boundary displacement is sufficiently large. We will prove

that such a discontinuous minimiser is a limit (as ε → 0) of a corresponding sequence

of minimisers of “regularised” problems in which the body contains a pre-existing hole of

radius ε (centred on x0 ) in its reference configuration (see [3]). These results involve use

of the Brouwer degree and an invertibility condition introduced by Muller and Spector [4].

Finally, using ideas from [2], we indicate possible applications of these results to modelling

the initiation of fracture.

References

1. J. Sivaloganathan and S. J. Spector, ”On the existence of minimisers with prescribed

singular points in nonlinear elasticity”, J. Elasticity 59 (2000), 83–113.

2. J. Sivaloganathan and S. J. Spector, ”On cavitation, configurational forces and impli-

cations for fracture in a nonlinearly elastic material”, J. Elasticity 67 (2002), 25–49.

3. J. Sivaloganathan, S.J. Spector and V. Tilakraj, ”The convergence of regularised

minimisers for cavitation problems in nonlinear elasticity”, (Preprint 2003).

4. S. Muller and S.J. Spector, ”An existence theory for nonlinear elasticity that allows

for Cavitation”, Arch. Rational Mech. Anal. 131 (1995), 1–66.

21

35 Finite indentation of an elastic fibre-reinforced sheet

A.J.M. Spencer, Department of Theoretical Mechanics, The University of Nottingham,

Nottingham NG7 2RD.

Email: anthony j [email protected]

In the forming of fibre- reinforced sheets, it is often observed that thinning of the sheet

occurs in regions of high normal pressure (which are usually areas in which the curvature

is large), with consequent spread of the fibres. As an initial contribution to the analysis

of this phenomenon, we consider the problem of finite indentation by normal pressure of a

flat sheet of initially unidirectionally reinforced elastic material of uniform thickness. The

model incorporates the kinematic constraints of incompressibility and fibre inextensibility.

The governing equations are hyperbolic, with the deformed fibre directions and their normal

trajectories as characteristics. If the deformed thickness is specified, then the problem is

kinematically determined and a numerical procedure is described which determines the

deformed fibre directions. However if normal pressure rather than displacement is specified

on part of or the entire surface, then it is necessary to take account of the material properties

through a constitutive equation for the stress. A finite elastic model is developed for

the considered class of deformations. In the case of specified pressure it is not possible

to separate the kinematic problem from the determination of the stress response, but an

iterative procedure is developed that leads to a complete solution. Test problems considered

are (a) a sheet indented by the curved surface of a circular cylinder lying oblique to the

fibres, and (b) a sheet indented by a sphere. An analytical solution to problem (a) is

obtained and is used to verify the numerical procedure.

36 Stability of localized buckling solutions for dead and rigid

loading in a model structure

M.K. Wadee, School of Engineering and Computer Science, University of Exeter, North

Park Road, Devon, EX4 4QF.

[email protected]

Localized (homoclinic) post-buckling solutions are known to be the preferred form of

deflection pattern for the model problem of an axially-compressed elastic strut resting on

a softening elastic foundation. Some stability results have previously been derived for

solutions which bifurcate from a Hamiltonian-Hopf bifurcation at least for a certain type

of nonlinearity. We apply a non-periodic Rayleigh-Ritz procedure and use basic arguments

about the potential energy of the structure to study the stability of localized solutions

under conditions of load- and displacement-control for a broader variety of nonlinearities

which, it may be argued, are more applicable to real structural problems. Comparisons

with published results is encouraging.

22

37 A fracture criterion of “Barenblatt” type for an intersonic

shear crack

J.R. Willis, Department of Applied Mathematics and Theoretical Physics, Centre for

Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA.


Steady-state intersonic propagation of a shear crack is considered, with the admission of

cohesion across the crack faces. The asymptotic limit of “small-scale cohesion”, which oc-

curs when the magnitude of the cohesive stress far exceeds that of the applied stress, is

developed explicitly, to obtain a criterion of “Barenblatt” type. The application of this

criterion requires only the calculation of the “applied” stress intensity coefficient with cohe-

sion disregarded; an equation of motion follows by equating this coefficient to a “modulus

of cohesion” which depends on the cohesive model that is employed. An explicit formula

for the “modulus of cohesion” is given for the special case of a cohesive zone of Dugdale

type.

23

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