2. J. Badur and Y. Povstenko, On two reinterpretations of Cosserat continuum: fiber bundle versus the motor calculus (abstract)

3. R. Gorenflo and F. Mainardi, Fractional calculus and stable probability distributions (abstract)

4. G. Hatfield and P.J. Olver, Canonical forms and conservation laws in linear elastostatics (abstract)

5. R. Ibáñez, M. de León, J.C. Marrero and D.M. de Diego, Nambu-- Poisson dynamics (abstract)

6. I. Kolar, On generalized parallelisms (abstract)

7. W. Kosiñski, Hyperbolic framework for thermoelastic materials (abstract)

8. E. Kroener, Random elastic media: why zero mean stress does not imply zero mean strain (abstract)

9. M. Kures, Some properties of connections on iterated tangent bundles (abstract)

10. T. Lipniacki, Dynamics of quantum vortices in superfluid ⁴He (abstract)

11. J. Makowski, W. Pietraszkiewicz and H. Stumpf, On the general form of jump conditions for thin irregular shells (abstract)

12. B. Maruszewski, Superconducting fullerenes in a nonconventional thermo-dynamical model (abstract)

13. G.A. Maugin, Thermomechanics of forces driving singular point sets (abstract)

14. G.A. Maugin and L. Restuccia, Pseudomomentum in relativistic continuum mechanics (abstract)

15. A.I. Murdoch, On effecting averages nad changes of scale via weighting functions (abstract)

16. W. Muschik, Objectivity and frame indifference, revisited (abstract)

17. M. Ostoja-Starzewski, Random field models and scaling laws of hetero\-genous media (abstract)

18. C. Papenfuss, Nonlinear dynamics of the alignment tensor in the presence of electric fields (abstract)

19. G.P. Parry, On defective crystallography (abstract)

20. M. Scholle, System symmetries and inverse variational problems in continuum theory (abstract)

21. M.L. Szwabowicz, A note on kinematics of surfaces (abstract)

22. C. Trimarco, The Maxwell rule in phase-transitions(abstract)

23. £.A. Turski, Diffusion (abstract)

24. H.J. Wagner, On generalized Weber and Clebsch transformations (abstract)

25. K. Wilmanski, On the time of existence of weak discontinuity waves in poro-elastic materials (abstract)

The following paper was included in the Proceedings but it was not presented during the meeting:

26. C. Sansour, A theory of the elastic-viscoplastic Cosserat continuum (abstract)

Within the Lagrange formalism, a mechanical continuum theory of dislocation dynamics is presented, which results in a phenomenological, unified description of elastic and plastic deformations of a crystal. Further developments towards a thermo-mechanical theory including dissipation are methodically envisaged. The theory is based on complex matter fields and vortex potentials as fundamental field variables. Especially the dislocation network is divided into different classes of equal dislocations, giving rise to a more refined description of the dislocation dynamics as traditionally can be done by the well-known dislocation density tensor. Each class of dislocations is associated with a complex dislocation field. The elastic interaction between dislocations of different classes results in correlational effects which cannot be described by means of the traditional continuum theory of dislocations. Whereas in traditional approaches the plastically deforming body is formally looked upon as an elastic solid with inherent flow properties, we are looking at such a system in a reverse manner: The plastically deforming body is formally regarded as a fluid with inherent solid properties. Formally the plastically deforming body is associated with a generalized Cosserat fluid based on matter and dislocation fields. In this way we overcome the difficulties due to the deformation chaos produced by dislocation motion.

It is the purpose of this paper to reinterprete the original Cosserat continuum from the point of view of both the fiber bundles geometry and the non-Abelian motor calculus. The main ideas of Cosserats are best explained in terms of the moving reper which is synonymous with the procedure of gauging in physics as well as with the procedure of constructing a fiber bundle in pure geometry. On the other hand, the classical (linear) von Mises motor calculus is extended to a non-Abelian case. It also appears that this non-Abelian version of the von Mises concept is fully equivalent with the fiber bundle description.

Fractional calculus allows one to generalize the linear (one-dimensional) diffusion equation by replacing either the first time-derivative or the second space-derivative by a derivative of a fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions. For the space fractional diffusion, a random-walk model is also proposed.

Several physical systems can be described by using multibrackets instead of the usual Poisson or Jacobi brackets. Starting with the original construction of Nambu we give a brief review on recent results on multibrackets on manifolds.

M. Epstein and M. de Leon defined the second order non-holonomic parallelism on a manifold and applied it to a geometric description of generalized Cosserat continua. We explain that the underlying geometric idea is the concept of generalized parallelism on an arbitrary principal fiber bundle. Some properties of generalized parallelisms are characterized in terms of induced connections or from the viewpoint of the theory of generalized G-structures.

A thermodynamic framework of a deformable continuum is developed in which the conservative state variable vector is enlarged by adding the spatial gradient of a scalar thermal internal state variable responsible for the description of thermal history effects. The theory leads to a modified model of thermoelasticity with internal state variables and with a wave-type heat conduction governed by a system of quasi-linear hyperbolic equations. In a general non-deformable case, the observed material properties such as specific heat, quasi-equilibrium thermal conductivity and speed of thermal (the so-called second sound) waves, all regarded as functions of t, lead to a particular specification of all material functions and the evolution equation for the scalar internal state variable. The short review of the heat pulse experiment is made. Main assumptions of the present approach are formulated and some arguments referred to the rate-type description are presented. A set of remarks and comparisons with another modification of the Fourier law characterizes the model. An analysis of hyperbolicity of thermoelasticity by propagation conditions of weak discontinuity waves is performed.

In this note we are concerned with (linearized) elastic media that are heterogeneous on the microscale and homogeneous on the macroscale. We assume the validity of an ergodic hypothesis so that we can form ensemble averages such as mean stress <T> and mean strain <e>. We argue that <T>=0 does not, in general, imply that also <e>=0. This is the case, for instance, when the distribution of the stress sources (external forces or incompatibilities) are correlated with the spatial distribution of the local elastic moduli. It is shown how problems of this type can be treated.

Possibilities of a generalization of the original Grifone's approach to connections are studied. Semisprays associated to connections and torsions on the iterated tangent bundle TTM are described.

The dissipative motion of quantized vortex line, after reconnection, is studied within the localized induction approximation. The numerical simulations of vortex line evolution help to determine the rate of the line-length changes. In the absence of counterflow, the vortex line shortens after each reconnection and line-length reduction is calculated as a function of friction parameter £ and reconnection angle "phi". The obtained results sugest that the decay of quantized vortex tangle is due to line-length reduction which occurs after each reconnection of vortex lines. In the presence of the counterflow, however, the reconnection may initiate a generation of a cascade of vortex loops. These loops blow up, so the total length of vortices grows up and the quantum turbulence can be generated.

The paper deals with the nonlinear theory of thin shell structures in the presence of irregularities in geometry, deformation, material properties and loading. The irregular shell is modelled by a reference network being a union of piecewise smooth surfaces and space curves, with various fields satisfying relaxed smoothness, differentiability, and regularity requirements. Transforming the virtual work principle postulated for the entire reference network, the corresponding local field equations and side conditions (boundary and jump conditions) are derived. It is shown that no more than four static and work-conjugate kinematic jump conditions can correctly be formulated whenever the shell deformation is assumed to be entirely determined by deformation of the reference network capable of resisting to stretching and bending. This assumption includes various special formulations of the Kirchhoff--Love type theory of elastic shells, as well as their substantial generalizations accounting for finite strains and inelastic deformations.

A crystalline solid of C60 (a fullerene) is expected to be an insulator or a semiconductor. However, one of the most striking properties of C60 -- related materials is the observation of relatively high temperature superconductivity in alkali metal doped M3C60 and in various alkaline earth doped compounds. So, the interstitial diffusion of impurities considerably influences the superconductivity phase. The value of the critical temperature below which the superconducting phase exists, strongly depends on many other external influences (electromagnetic, thermal, mechanical, etc.). The paper deals with construction of a phenomenological macroscopic model of interactions between physical fields in fullerenes, basing on the extended thermodynamics with internal variables and with the use of Liu's theorem in order to apply the entropy inequality.

By treating in parallel the balance of canonical momentum and the entropy equation, both at regular material points and at singular sets such as discontinuity fronts, it is shown that a consistent thermomechanics of such fronts can be constructed, especially with regard to shock waves and phase-transition fronts. Within this framework, two extreme singular cases are that of the classical shock-wave theory which relates dissipatively two states in adiabatic evolution, and that of the nondissipative phase transition which relates two generally dissipative states. In both cases, the driving force on the singular set is made to vanish yielding oversimplifications. This is obviously corrected by showing that if dissipation occurs at all, such a driving force should not be zero. It is in fact related to the details of what happens within a structured front and to the noninertial motion of such a front viewed as a quasi-particle. In passing, the role of a generating (thermodynamic) function for discontinuity fronts is exhibited.

In classical continuum mechanics the balance or unbalance equation of pseudomomentum reflects the material invariance of the system under study. It relates the time derivative of pseudo-momentum and the flux of the Eshelby stress. It is legitimate to inquire whether this structure is conserved in a relativistic four-dimensional background. We examine here the relativistic definition of pseudo/material momentum using simultaneously variational and direct approaches (the latter using the canonical projection of as space-time onto the material manifold). It appears that the truly {\it material entities}, just as those in a proper frame, should be the basic ones, being independent of the relativity framework used.

Weighting functions can be used to derive continuum equations of balance from molecular considerations, and to obtain equations governing fluid flow through porous media. The methodology of such (scale-dependent) averaging is outlined, and physical implications of specific choices of weighting function are discussed.

Because one has to distinguish between changing the observer and changing the state of motion of materials, objectivity and material frame indifference are redefined: Objectivity denotes a special tensor property in case of changing the observer, whereas material frame indifference is characterized by quantities being independent of different states of motion of the material. To describe these different states of motion of the material, an arbitrary standard frame of reference and a Constitutive Family is introduced. We prove that the constitutive map is isotropic in the state variables, if these and the material properties are objective.

In many problems of solid mechanics (e.g., stochastic finite elements, statistical fracture mechanics) there is a need for resolution of dependent fields over scales not infinitely larger than the microscale. This task may be accomplished through a ``meso-scale window" which becomes the classical Representative Volume Element (RVE) in the infinite limit relative to the microscale. It turns out that the material properties at such a mesoscale cannot be uniquely approximated by a random field of stiffness/compliance with locally isotropic realizations, but, rather, two random continuum fields with locally anisotropic realizations, corresponding respectively to Dirichlet and Neumann boundary conditions on the meso-scale, need to be introduced to bound the material response from above and from below. We discuss statistical characteristics of these two mesoscale random fields, including their spatial correlation structure, for anti-plane elastic response of random two-phase composites with Voronoi geometry at the percolation point. Particular attention is given to the scaling of effective responses obtained from both conditions, which sheds light on the minimum acceptable size of an RVE.

In the mesoscopic theory field quantities are introduced, which depend not only on position and time, but also on an additional orientational variable, the microscopic director. The orientation distribution function (ODF) gives the fraction of particles of a particular orientation. An equation of motion for the second order alignment tensor, including the influence of electromagnetic fields and of spatial inhomogeneities, is derived. The starting point are the mesoscopic balance equations of mass and of angular momentum. A constitutive equation on the mesoscopic level is discussed. The two balance equations together with the constitutive equation yield an equation of motion for the ODF. This finally leads to a nonlinear partial differential equation for the alignment tensor, which involves also the fourth and the sixth moment of the ODF.

 
 

Suppose that a solid crystal derives from a perfect (Bravais) lattice of atoms, so that the set of rearrangements of the points of this lattice provides symmetries of the crystal. In the elasticity theory appropriate to such a crystal, it is traditional to assert that the corresponding strain energy density function has invariance properties related to a (proper) subset of these symmetries. Here I discuss similar issues in the context of the continuum mechanics of smoothly defective crystals, focusing on planar distributions of defects.

The aim of the conventional Inverse Problem in Lagrange formalism is to find a Lagrangian, the associated Euler--Lagrange equations of which are equivalent to a given set of partial differential equations of a physical system. In contrast, I am dealing with a different type of an inverse problem. I look for a Lagrangian which is associated with a given set of balance equations. My approach is based on general relations between symmetry groups (geometrical and gauge symmetries) and its associated balance equations. I follow two different mathematical lines: The first one is Noether's theorem: Universal Lie symmetry groups like translations (spatial and temporal), rotations and Galilei transformation are connected with the fundamental conservation laws for energy, linear momentum, angular momentum and center of mass motion. All of these balances are of the ``volume-type". The second line takes account of a relationship between non-Lie symmetry groups (e.g. regauging of potentials) and balances of the ``area-type". These are physically associated with line-shaped objects like vortex lines and dislocations. Following both lines in an inverse manner I derive the relevant symmetry properties of a yet unknown Lagrangian for a given set of balance equations of volume- and area-types. Consequently, a rough scheme for the analytical structure of the Lagrangian can be given. As an example, a Lagrangian for the elastic deformation of a body with eigenstresses due to fixed dislocations is constructed.

The question how to describe effectively the motion of a deformable surface in ordinary Euclidean space is discussed. Two alternative formulations are supplied, both based on the assumption that the Riemannian metric of the moving surface must appear explicitly in the system describing the motion. The motivation for this assumption is to divide the variables responsible for the evolution of the intrinsic geometry (strains) of the surface from those responsible for the evolution of its extrinsic geometry (bending). Exemplary application of these results to the large deflection/small strain class of deformations of thin shells is considered.

The Maxwell rule, also known as the rule of equal areas, represents a basic notion in the phase transition processes. Although the rule might need amending for solids, it is treated as unquestionable for fluids. This belief relies on a thermodynamical argument. Namely, that the Gibbs thermodynamical potential should remain stationary during the transition process. Surprisingly, from the same thermodynamical argument, the Maxwell rule can be invalidated, even for fluids. Nonetheless, a revised form of the rule can be proposed.

The classical field of statistical mechanics -- the theory of diffusion processes -- is still offering considerable challenge when the physical problems to be described by it are more ``realistic'' than those easily envisioned as simple random walk. In this lecture I shall present our recent results on diffusion processes in two wide classes of physical problems: i) Diffusion in dense quasi-two-dimensional adsorbates on surfaces of the crystals, where the interparticle interactions and the interaction with the host solid cannot be neglected and play a mutually complementary role. These phenomena can be conveniently called the dynamics in d=2+1 dimensions. ii) Diffusion in the crystals containing topological (line) defects, such as dislocations and disclinations. I shall present our results on use of the combined continuum theory of defects and the path integral approach to description of such diffusion processes. Possibility of generalization of these models for quantum particles will also be outlined.

Suitable generalizations of the Weber and Clebsch transformations of the hydrodynamic equations are introduced which have some bearing in the treatment of the inverse problem of Lagrangian field theory. In particular these generalizations open the way to equivalence proofs for several Lagrangians proposed in the realm of ideal (magneto-) hydrodynamics. This means that the Euler--Lagrange equations corresponding to these Lagrangians do not only imply but are also implied by the original field equations of the systems under study.

In the paper, we consider the possibility of the growth of strong discontinuity waves in the two-component poroelastic materials. We use the model with the hyperbolic set of field equations described in the paper by K. Wilmañski [8]. It is shown that indeed, the critical time (i.e. the maximum time of existence of classical solutions) is finite and it assumes realistic values for real physical systems, such as biological tissues.

Based on the multiplicative decomposition of the stretch tensor and the additive decomposition of the second Cosserat deformation tensor into elastic and inelastic parts, a theory of the elastic-viscoplastic Cosserat continuum is formulated. It is stressed that the rotation field is to be treated as a kinematical variable which can not be decomposed into elastic and inelastic parts. A thorough discussion of the configuration space by relying on basic concepts of Lie groups is provided and the field equations are derived from a variational statement. The flow rules are specified by means of the postulate of maximum dissipation paralleling some developments of the classical theory.