Showing papers in "Continuum Mechanics and Thermodynamics in 2001"

A solid-fluid mixture model allowing for solid dilatation under external pressure

Abstract: A sponge subjected to an increase of the outside fluid pressure expands its volume but nearly mantains its true density and thus gives way to an increase of the interstitial volume This behaviour, not yet properly described by solid-fluid mixture theories, is studied here by using the Principle of Virtual Power with the most simple dependence of the free energy as a function of the partial apparent densities of the solid and the fluid The model is capable of accounting for the above mentioned dilatational behaviour, but in order to isolate its essential features more clearly we compromise on the other aspects of deformation Specifically, the following questions are addressed: (i) The boundary pressure is divided between the solid and fluid pressures with a dividing coefficient which depends on the constituent apparent densities regarded as state parameters The work performed by these tractions should vanish in any cyclic process over this parameter space This condition severely restricts the permissible constitutive relations for the dividing coefficient, which results to be characterized by a single material parameter (ii) A stability analysis is performed for homogeneous, pressurized reference states of the mixture by postulating a quadratic form for the free energy and using the afore mentioned permissible constitutive relations It is shown that such reference states become always unstable if only the external pressure is sufficiently large, but the exact value depends on the interaction terms in the free energy The larger this interaction is, the smaller will be the critical (smallest unstable) external pressure (iii) It will be shown that within the stable regime of behaviour an increase of the external pressure will lead to a decrease of the solid density and correspondingly an increase of the specific volume, thus proving the wanted dilatation effects (iv) We close by presenting a formulation of mixture theory involving second gradients of the displacement as a further deformation measure (Germain 1973); this allows for the regularization of the otherwise singular boundary effects (dell'Isola and Hutter 1998, dell'Isola, Hutter and Guarascio 1999)

Stress–strain relations in finite viscoelastoplasticity of rigid-rod networks: applications to the Mullins effect

Abstract: Constitutive equations are derived for the time-dependent behavior of a network of rigidrod chains at finite strains. The stress‐strain relations are applied to describe the Mullins effect: a noticeable difference between the response of elastomers under tension and subsequent retraction during the first cycle of periodic loading. Adjustable parameters in the model are determined by fitting observations for several grades of particle-reinforced rubber at uniaxial elongation (up to 300%) with subsequent retraction and at uniaxial extension with the maximal strain up to 600%. Fair agreement is demonstrated between experimental data and results of numerical simulation. It is revealed that material constants alter with the filler concentration in a physically plausible way and demonstrate dramatic changes at the percolation threshold corresponding to transition from isolated clusters of particles to a filler network.

Decomposition and integral representation of Cauchy interactions associated with measures

Abstract: Cauchy interactions between subbodies of a continuous body are introduced in the framework of Measure Theory, extending the class of previously admissible ones. A decomposition theorem into a volume and a surface interaction is proved, as well as characterizations of the single components. Finally, an extension result and a generalized balance law are given.

Dispersion and absorption of plane harmonic waves in a relativistic gas

Abstract: The dispersion and absorption of plane harmonic waves in a relativistic gas which is described by a five-field theory with Navier-Stokes-Fourier and Burnett constitutive equations and by a thirteenand fourteen-field theories are analyzed. In all cases the constitutive equations are derived from the Boltzmann equation where a constant differential cross-section is considered. The differences between the expressions for the dispersion and absorption of the plane waves that follow from the five-field theories with Navier-Stokes-Fourier and Burnett constitutive equations and from the thirteenand fourteen-field theories in the ultra-relativistic and in the non-relativistic limits are given.

Instabilities in an open top horizontal porous layer subjected to pulsating thermal boundary conditions

Abstract: Consider a horizontal porous layer, saturated by a fluid which is free to cross the upper boundary. Suppose, in addition, that the temperature of the upper boundary undergoes sinusoidal fluctuations. The filtration speed and the temperature then satisfy a nonlinear system of partial differential equations which admits a quiescent solution. Within the frame-work of the Darcy law, three parameters appear in the system governing the evolution of the temperature. They are the amplitude and the non-dimensional frequency of the thermal forcing, in addition to the average vertical thermal gradient. Two questions are addressed. For given frequency the limit of unconditional stability of the quiescent state is studied: when the amplitude of the forcing is under the limit, every starting motion is monotonically damped. Above, some perturbations amplify at least for a while. Another threshold is characterized for the possible existence of permanent seeping currents. The results hold for the full nonlinear problem, without any restriction connected with the amplitude of the motion. Nevertheless the compound matrix method gives the thresholds directly. The kinematic boundary conditions, prescribed on the surface where the thermal fluctuations are enforced, help to destabilize the quiescent state.

Forces on nematic disclinations with optimal core

Abstract: =+1 within a capillary tube, reducing the problem to a planar one. We show that, if the disclination is not close to the capillary boundary, its core has nearly circular cross-sections that are not coaxial with the disclination. On the contrary, when the disclination is close to the capillary boundary its core suffers large distortions from a circular shape. The force acting on a disclination, that dictates its incipient dynamics, is computed as a function of its distance from the capillary axis, and is compared to that obtained when the core cannot change its shape.

On geometric acoustics in random, locally anisotropic media

Abstract: When considering spatially inhomogeneous media with smooth variability, it is natural to admit local anisotropy – a non-spherical indicatrix – in place of a more restrictive assumption of isotropy. On that basis, ray dynamics and the eikonal equation that govern geometric optics (and geometric acoustics) in inhomogeneous media are formulated to take into account the dependence of local propagation velocity on the direction of the ray. All the equations reduce to classical forms when anisotropy vanishes.

Spatial decay estimates for heat-conducting fluids

Abstract: This paper derives spatial decay bounds for the linear dynamical problem of a heat conducting fluid, defined on a semi-infinite cylindrical region. Previous results concerning parabolic equations lead us to suspect that the solution of the problem should tend to zero faster than a decaying exponential of the distance from the finite end of the cylinder. We prove that an energy expression is actually bounded above by a decaying exponential of a quadratic polynomial of the distance.

Constitutive equations of extended thermodynamics from a hybrid pair of generator functions

Abstract: Exploitation of constitutive restrictions in extended thermodynamics rests mainly either on representations of constitutive functions in terms of the basic elds or representations of generator functions in terms of the Lagrange multipliers (the main elds). The latter is more systematic and elegant, however, in order to replace the Lagrange multipliers with the basic elds, an inversion of a generally nonlinear system of equations is involved. Usually only the linear case can be inverted easily. A dieren t approach to avoid such inversions is proposed, which employs representations of a hybrid pair of generator functions in terms of the basic elds and the main elds as variables respectively. This procedure is illustrated in deriving the constitutive equations of ideal gases with 14 moments. The domain of the approximated constitutive functions, limited by the concavity requirement of the entropy function, is also considered.

Generalized strain and stress tensors associated by duality

Abstract: We consider the generalized Lagrangean strain measures introduced previously by Hill e.g. [1,2]. By requiring that some scalar quantities be form-invariant, generalized strain and dual stress tensors as well as associated rates have been derived by Haupt & Tsakmakis [3]. In the present work we discuss some mathematical aspects of the duality concept of Haupt & Tsakmakis [3] from a local geometric point of view.

Steady plane nonlinearly viscous flow of ice sheets on bedswith moderate slope topography

Abstract: , the aspect ratio of the ice sheet, \(\delta\), the maximum bed slope (\(\epsilon \ll \delta \ll 1\)), and one of either a, the maximum amplitude of the topography, or s, the span over which local topography extends. For steady plane flow of nonlinearly viscous ice the form of the leading order and two correction terms in the asymptotic expansion is deduced and the boundary value problems for the leading order and first correction terms are presented. As illustrations, the leading order and correction terms of the horizontal velocity are calculated and plotted for typical bed configurations. Comparison are then made with the three simpler cases of ice response: isothermal and nonlinearly viscous, then temperature dependent and linearly viscous, and finally isothermal and linearly viscous. It is found that for all three the correction terms in the horizontal velocity are very small.

Stability results for convection in thawing subsea permafrost

Abstract: unconditional nonlinear stability.