The effect of fiber recruitment on the swelling of a pressurized anisotropic non-linearly elastic tube

We study the effect of fiber recruitment on the mechanical response of a fiber reinforced non-linearly elastic tube that is both swollen and pressurized. Attention is restricted to cylindrically symmetric tube deformation. The constitutive model permits fibers to support tension, but not compression. While many combinations of pressure and swelling cause all of the fibers to be recruited for load support, both large swelling and large deswelling can give rise to fiber derecruitment at certain locations in the tube. This leads to less channel opening than would be the case if the fibers provided support while contracted. The transition between mechanically active and mechanically inactive fibers can be described in terms of the quasi-static motion of a fiber recruiting interface.     

Multidimensional stability of martensite twins under regular kinetics

This paper considers phase boundaries governed by regular kinetic relations as first proposed by Abeyaratne and Knowles [1990. On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids 38 (3), 345-360; 1991. Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119-154]. It shows that static configurations of hyperelastic materials, in which two different martensitic (monoclinic) states meet along a planar interface, are dynamically stable towards fully three-dimensional perturbations. For that purpose, the reduced stability (or reduced Lopatinski) function associated to the static twin [Freistühler and Plaza, 2007. Normal modes and nonlinear stability behavior of dynamic phase boundaries in elastic materials. Arch. Ration. Mech. Anal. 186 (1), 1-24] is computed numerically. The results show that the interface is weakly stable under Maxwellian kinetics expressing conservation of energy across the boundary, whereas it is uniformly stable with respect to linearly dissipative kinetic rules of Abeyaratne and Knowles type.     

On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem

We consider here a model of fluid-structure evolution problem which, in particular, has been largely studied from the numerical point of view. We prove the existence of a strong solution to this problem.     

Singular limit of equations for linear viscoelastic fluids with periodic boundary conditions

We consider a one-parameter family of problems, governing, for any fixed parameter, the motion of a linear viscoelastic fluid in a two-dimensional domain with periodic boundary conditions. The asymptotic behavior of each problem is analyzed, by proving the existence of the global attractor. Moreover, letting the parameter go to zero, since the memory effect disappears, we obtain a limiting problem, given by the Navier-Stokes equations. For any fixed parameter, we construct an exponential attractor. The resulting family is robust, meaning that these exponential attractors converge, in an appropriate sense, to an exponential attractor of the limiting problem.     

On Compactness, Domain Dependence and Existence of Steady State Solutions to Compressible Isothermal Navier–Stokes Equations

The steady state system of isothermal Navier–Stokes equations is considered in two dimensional domain including an obstacle. The shape optimisation problem of minimisation of the drag with respect to the admissible shape of the obstacle is defined. The generalized solutions for the Navier–Stokes equations are introduced. The existence of an optimal shape is proved in the class of admissible domains. In general the solutions are not unique for the problem under considerations.     

A New Derivation of Jeffery’s Equation

In this article, we present a modern derivation of Jeffery’s equation for the motion of a small rigid body immersed in a Navier–Stokes flow, using methods of asymptotic analysis. While Jeffery’s result represents the leading order equations of a singularly perturbed flow problem involving ellipsoidal bodies, our formulation is for bodies of general shape and we also derive the equations of the next relevant order.      

On the Averaging of Symmetric Positive-Definite Tensors

In this paper we present properly invariant averaging procedures for symmetric positive-definite tensors which are based on different measures of nearness of symmetric positive-definite tensors. These procedures intrinsically account for the positive-definite property of the tensors to be averaged. They are independent of the coordinate system, preserve material symmetries, and more importantly, they are invariant under inversion. The results of these averaging methods are compared with the results of other methods including that proposed by Cowin and Yang (J. of Elasticity 46 (1997) pp. 151–180.) for the case of the elasticity tensor of generalized Hooke's law.     

Derivation and Justification of the Models of Rods and Plates From Linearized Three-Dimensional Micropolar Elasticity

In this paper we derive and mathematically justify models of micropolar rods and plates from the equations of linearized micropolar elasticity. Derivation is based on the asymptotic techniques with respect to the small parameter being the thickness of the elastic body we consider. Justification of the models is obtained through the convergence result for the displacement and microrotation fields when the thickness tends to zero. The limiting microrotation is then related to the macrorotation of the cross–section (transversal segment) and the model is rewritten in terms of macroscopic unknowns. The obtained models are recognized as being either the Reissner–Mindlin plate or the Timoshenko beam type.     

A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data

We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|_∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.     

Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid

We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large. The existence result presented here has been announced in [4]. Some improvements have been provided on the model: the model considered in [4] is a simplified model where the structure motion is modelled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. In what follows, we consider on the structure a model which represents the motion of a structure with large rigid displacements and small elastic perturbations. This model, introduced by [15] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.     

Some Basis-Free Formulae for the Time Rate and Conjugate Stress of Logarithmic Strain Tensor

In this paper, two kinds of tensor equations are studied and their solutions are derived in general cases. Then, some compact basis-free representations for the time rate and conjugate stress of logarithmic strain tensors are proposed using six different methods. In addition, relations between the coefficients in these expressions are disclosed. Subsequently, all these basis-free expressions given in this paper are validated for the cases of distinct stretches and double coalescence, respectively.     

Euler–Lagrange/DEM simulation of wood gasification in a bubbling fluidized bed reactor

We present an Euler–Lagrange method for the simulation of wood gasification in a bubbling fluidized bed. The gas phase is modeled as a continuum using the 2D Navier–Stokes equations and the solid phase is modeled by a Discrete Element Method (DEM) using a soft-sphere approach for the particle collision dynamic. Turbulence is included via a Large-Eddy approach using the Smagorinsky sub-grid model. The model takes into account detailed gas phase chemistry, zero-dimensional modeling of the pyrolysis and gasification of each individual particle, particle shrinkage, and heat and mass transfer between the gas phase and the particulate phase. We investigate the influence of wood feeding rate and compare exhaust gas compositions and temperature results obtained with the model against experimental data of a laboratory scale bubbling fluidized bed reactor.     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

The purpose of this work is to study the existence of solutions for an unsteady fluid-structure interaction problem. We consider a three-dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with a flexible elastic plate located on one part of the fluid boundary. The fluid domain evolves according to the structure’s displacement, itself resulting from the fluid force. We prove the existence of at least one weak solution as long as the structure does not touch the fixed part of the fluid boundary. The same result holds also for a two-dimensional fluid interacting with a one-dimensional membrane.     

An Evolution Principle and the Existence of Entropy. First Order Systems

We formulate a basic principle, called evolution principle, for a given set of physical processes. Then we consider a set given by solutions of first order systems without reaction terms. We show that for strictly hyperbolic systems and for the Euler system the evolution principle is equivalent to the entropy principle. Received May 20, 1997     

Fast Decaying Solutions of the Navier-Stokes Equation and Asymptotic Properties

While the basic global existence problem for the Navier-Stokes equations seems to remain open, there are related questions of some interest which are amenable to discussion: find large initial data giving rise to global solutions. Such initial data are known in the literature. A study shows that they have a peculiar property: they give rise to solutions which decay fast in very short time. A major result to be proved states that the set of trajectories induced by such initial data is dense in every open set (with respect to some fractional power norm). A further result states that if the exterior force f is zero, then such rapid decays cannot occur infinitely often along trajectories. This follows from some inequalities, connecting and , with A the Stokes operator.     

An Evolution Principle and the Existence of Entropy. ODE Case

We formulate a basic principle, called evolution principle, and show that in the case of an underdetermined ODE it is equivalent to other existing principles, which are the entropy principle and Carathéodory's inaccessibility axiom. In a forthcoming paper we shall formulate the same principle in the PDE case. Received May 20, 1997