modern mechanics and mathematics – an international conference in honour of ray ogden’s 60th...
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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.
Email: [email protected]
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.
Email: [email protected]
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]
Email: [email protected], and [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
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
5. A.B. Freidin and A.M. Chiskis, Phase transition zones in nonlinear elastic isotropic
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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.
Email: [email protected]
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
Email: [email protected], and [email protected]
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
1. 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) 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.
Email: [email protected]
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.
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.
Email: [email protected]
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