
Plenary
Lecture
Spatial Behaviour in Continuum Mechanics
Professor Stan
Chirita
Faculty of Mathematics
Al. I. Cuza University of Iasi
Blvd. Carol I, no. 11, 700506
Iasi, Romania
Email:
schirita@uaic.ro
Abstract: In the construction of buildings, bridges,
aircraft, nuclear reactors and automobiles, the engineer
must determine the depth to which local stresses, such
as those produced by fasteners and at joints, or
vibrations can penetrate girders, Ibeams, braces and
other similar structural elements. The determination of
the extent of local or edge effects in structural
systems allows the engineer to have a clear distinction
between the global structure (where strength of
materials approximations can be used) and the local
excited portions which require a separate and more
elaborate analysis based on some exact theories as that
of linear elasticity. The standard procedure used in
engineering practice to determine the extent of local
stresses or edge effects is based on some form of the
celebrated Saint Venant principle. A comprehensive
surveys of contemporary research concerning Saint Venant
principle can be found in Horgan and Knowles (1983) and
Horgan (1989; 1996).
As regards elastic vibrations, it was observed in these
papers that high frequency effects might be expected to
propagate with little spatial attenuation (see also
Boley (1955; 1960)). It is outlined in Horgan and
Knowles (1983) that one would not expect to find
unqualified decay estimates of the kind concerning
SaintVenant's principle in problems involving elastic
wave propagation, even if the end loads are
selfequilibrated at each instant. In this connection,
Flavin and Knops (1987) have carried out an analysis of
spatial decay for certain damped acoustic and
elastodynamic problems in the low frequency range which
substantiates the early work of Boley. These results are
extended to linear anisotropic materials in Flavin et
al. (1990). It should be noted that all of the
investigations mentioned in the foregoing were concerned
with elastic materials having a positive definite
elasticity tensor.
On the other hand, in the literature concerning thermal
effects in continuum mechanics there are developed
several parabolic and hyperbolic theories for describing
the heat conduction. The hyperbolic theories are also
called theories of second sound and there the flow of
heat is modelled with finite propagation speed, in
contrast to the classical model based on the Fourier's
law leading to infinite propagation speed of heat
signals. A review of these theories is presented in the
articles by Chandrasekharaiah (1998) and Hetnarski and
Ignaczak (1999, 2000).
A new thermoelastic theory without energy dissipation
has been proposed by Green and Naghdi (1993). This
thermomechanical theory of deformable media introduces
the socalled thermal displacement relating the common
temperature and uses a general entropy balance as
postulated in Green and Naghdi (1977). By the procedure
of Green and Naghdi (1995), the reduced energy equation
is regarded as an identity for all thermodynamical
processes and places some restrictions on the functional
forms of the dependent constitutive variables. The
theory is illustrated in detail in the context of flow
of heat in a rigid solid, with particular reference to
the propagation of thermal waves at finite speed. The
linearized formulation allows the transmission of heat
flow as thermal waves at finite speed and the evolution
equations are fully hyperbolic.
The linear theory of thermoelasticity without energy
dissipation for homogeneous and isotropic materials was
employed by Nappa (1998) and Quintanilla (1999) in order
to obtain spatial energy bounds and decay estimates for
the transient solutions in connection with the problem
in which a thermoelastic body is deformed subject to
boundary and initial data and body supplies having a
compact support, provided positive definiteness
assumptions are supposed upon the constitutive coe±cients.
Moreover, we have to mention that Chandrasekharaiah
(1996) proves uniqueness of solutions, Iesan (1998)
establishes continuous dependence results, while
Quintanilla (2002) studies the question of existence.
Further results of structural stability and decay type
are given by Quintanilla (2001, 2003). Quintanilla and
Straughan (2000) used logarithmic convexity and Lagrange
identity arguments to yield uniqueness and growth
without requiring sign definiteness of the constitutive
coefficients, while Quintanilla and Straughan (2005)
derive energy bounds for a class of nonstandard
problems in which the initial data are given as a
combination of data at initial time and at a later time.
In the present lecture we address the question of
spatial behaviour of the harmonic vibrations and
transient solutions in an anisotropic elastic cylinder
under the condition of strong ellipticity for the
elasticity tensor. In this respect, for vibrations in
the low frequency range, our expected results describe
exponential spatial estimates similar with those
previously established by Flavin et al. (1987; 1990).
Moreover, for harmonic vibrations with appropriate high
frequencies, the present results predict some algebraic
spatial estimates, con?rming the foregoing observations
made by Boley in related context. In fact, we consider a
prismatic cylinder occupied by an anisotropic linear
elastic material and subjected to zero body force and
zero lateral boundary data and zero initial conditions.
The motion is induced by a harmonic time{dependent
displacement speci?ed pointwise over the base and the
other end is subjected to zero displacement (when a
cylinder of finite extent is considered, to say). The
elasticity tensor is assumed to be strongly elliptic and
so a very large class of anisotropic elastic materials
is considered, including those new materials with
extreme and unusual physical properties like negative
Poisson's ratio (that is, so called auxetic materials).
We also address the study of the spatial behaviour of
the transient and harmonic in time solutions for the
initial and boundary value problems associated with the
linear thermoelasticity theory without dissipation
energy for anisotropic materials. We derive some
differential inequalities for certain cross sectional
integrals and integration leads to estimates describing
how these integrals evolve with respect to the axial
variable. The methods employed, whose antecedent is the
technique developed by Flavin et al. (1990) for the
classical elastic problem and later developed Chirita
and Quintanilla (1996) and Chirita and Ciarletta (1999)
(for dynamic problems) and Chirita (1995) (for steady
state solutions), establishes differential inequalities
for the selected measures which after integration
provides estimates for spatial evolution, provided the
strong ellipticity of the constitutive coefficients is
assumed. However, here we use an idea developed by
Chirita (2007) for linear thermoelasticity of
anisotropic materials with a strong elliptic elasticity
tensor.
We examine how the amplitude of the harmonic vibration
evolves with respect to the axial variable. To this end
we associate with the amplitude of the harmonic
vibration in concern, an appropriate cross{sectional
integral function and further we prove that the strong
ellipticity conditions assure that it is an acceptable
measure. This is possible thanks to some appropriate
auxiliary identities relating the amplitude of the
harmonic vibrations. For these measures we are able to
establish some differential inequalities whose
integration allows us to obtain spatial estimates
describing the spatial behavior of the amplitude in
concern. In fact, when an identity of conservation
energy type is used then certain exponential spatial
estimates are obtained for all frequencies lower than a
critical value. When a Rellich identity is involved then
certain type of algebraic spatial estimates are
established for appropriate high frequencies. All
results are illustrated for transversely isotropic
materials as well as for the rhombic systems.
A description is also given for viscoelastic cylinders,
where the existence of the dissipation energy assures
the information upon the spatial behaviour of the
harmonic vibrations, without any restrictions upon the
relaxation tensor.
Our contribution lecture gives a complete discussion
about the state of art in the literature of spatial
behaviour of the transient and the harmonic in time
solutions based upon the recent results obtained by the
author and his collaborators and the articles in the
field of research on the subject in concern.
Brief Biography of the Speaker:
Stan Chirita was born on 21th October 1949 in Vizireni,
Buzau District, Romania. He graduated in 1972 from
Department of Mathematics at the Al.I. Cuza University
of Iasi, Romania. In 1977 he received the Doctor degree
in mathematics from Al. I. Cuza University of Iasi with
a thesis on the linear problems of the nonlocal theory
of elasticity. In 1987 he was awarded with the Gheorghe
Lazar Prize by the Romanian Academy for research results
on the nonlinear problems of continuum mechanics. He
published over 100 articles on the following fields of
research: elasticity, thermoelasticity, viscoelasticity,
NavierStokes fluids, generalized models of continuum
media, generalized theories of thermoelasticity,
nonsimple materials.
He was invited as Visiting Professor by the following
universities: University of Naples, University of
Bologna, University of Salerno and University of Catania
from Italy, University of Plymouth from England,
University of Barcelona from Spain, Ecole Centrale de
Lyon from France.
In the theory of elasticity he obtained results on the
following topics: spatial behaviour, SaintVenant
principle, SaintVenant problem, Deformation of
noncylindrical beams, Holder continuous dependence,
Thermal stresses. In the field of the classical
thermoelasticity I studied the following topics:
uniqueness and continuous data dependence problems,
asymptotic partition of energy, spatial behaviour,
backward in time thermoelasticity, Holder stability,
plate theory. In the field of materials with memory I
studied the following topics: uniqueness and continuous
data dependence, deformation of vicoelastic cylinders,
Saint—Venant principle, spatial behaviour, reciprocal
relations, thermoviscoelasticity, Reissner—Mindlin type
viscoelastic plate, thermodynamics of materials with
heat conduction and viscosity. In the field of fluids I
studied the following topics: spatial behaviour in
time—dependent Stokes slow flow, uniqueness and
continuous dependence problems for incompressible
micropolar flows forward and backward in time. In the
field of the generalized models of solids I studied the
following topics: nonlocal elasticity, micropolar
elasticity, materials with microstructure, materials
with voids, theory of mixtures. The following
generalized theories of classical thermoelasticity are
studied: the Green—Lindsay theory of thermoelasticity,
the theory of thermoelasticity with one relaxation time,
the linear thermoelasticity with memory for heat flux.

