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

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, I-beams, 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 Saint-Venant's principle in problems involving elastic wave propagation, even if the end loads are self-equilibrated 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 so-called 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 non-standard 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, Navier-Stokes fluids, generalized models of continuum media, generalized theories of thermoelasticity, non-simple 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, Saint-Venant principle, Saint-Venant 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.




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