A planar rod model with flexible thin-walled cross-sections. Application to the folding of tape springs

This paper is focused on the modeling of rod-like elastic bodies that have an initially curved and thin-walled cross-section and that undergo important localized changes of the cross-section shape. The typical example is the folding of a carpenter’s tape measure for which the folds are caused by the flattening of the cross-section in some localized areas. In this context, we propose a planar rod model that accounts for large displacements and large rotations in dynamics. Starting from a classical shell model, the main additional assumption consists in introducing an elastica kinematics to describe the large changes of the cross-section shape with very few parameters. The expressions of the strain and kinetic energies are derived by performing an analytical integration over the section. The Hamilton principle is directly introduced in a suitable finite element software to solve the problem. The folding, coiling and deployment of a tape spring is studied to demonstrate the ability of the model to account for several phenomena: creation of a single fold and associated snap-through behavior, splitting of a fold into two, motion of a fold along the tape during a dynamic deployment, scenarios of coiling and uncoiling of a bistable tape spring. This 1D model may also be relevant for future applications in biomechanics, biophysics and nanomechanics.     

Nonlinear Weakly Curved Rod by Γ-Convergence

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use an approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity. We do not impose any constitutional behavior of the material and work in a general framework. To derive the model, by means of ??-convergence, we need to set the order of strain energy (i.e., its relation to the thickness of the body h). We analyze the situation when the strain energy (divided by the order of volume) is of the order h ⁴. This is the same approach as the one used in F?ppl-von Kármán model for plates and the analogous model for rods. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.     

Dynamic symmetry breaking and structure-preserving analysis on the longitudinal wave in an elastic rod with a variable cross-section

The longitudinal wave propagating in an elastic rod with a variable cross-section owns wide engineering background, in which the longitudinal wave dissipation determines some important performances of the slender structure. To reproduce the longitudinal wave dissipation effects on an elastic rod with a variable cross-section, a structure-preserving approach is developed based on the dynamic symmetry breaking theory. For the dynamic model controlling the longitudinal wave propagating in the elast...     

Helical Birods: An Elastic Model of Helically Wound Double-Stranded Rods

We consider a geometrically accurate model for a helically wound rope constructed from two intertwined elastic rods. The line of contact has an arbitrary smooth shape which is obtained under the action of an arbitrary set of applied forces and moments. We discuss the general form the theory should take along with an insight into the necessary geometric or constitutive laws which must be detailed in order for the system to be complete. This includes a number of contact laws for the interaction of the two rods, in order to fit various relevant physical scenarios. This discussion also extends to the boundary and how this composite system can be acted upon by a single moment and force pair. A second strand of inquiry concerns the linear response of an initially helical rope to an arbitrary set of forces and moments. In particular we show that if the rope has the dimensions assumed of a rod in the Kirchhoff rod theory then it can be accurately treated as an isotropic inextensible elastic rod. An important consideration in this demonstration is the possible effect of varying the geometric boundary constraints; it is shown the effect of this choice becomes negligible in this limit in which the rope has dimensions similar to those of a Kirchhoff rod. Finally we derive the bending and twisting coefficients of this effective rod.     

Cataracting-centrifuging transition investigation using nonspherical and spherical particles in a rotary drum through CFD simulations

This paper aims to systematically investigate the cataracting-centrifuging transition in a rotary drum involving spherical and nonspherical particles by using the Multiphase Granular Eulerian Model (MGEM). The effects of drum length and particle shape on the cataracting-centrifuging transition behavior were analyzed. The results showed that drum length plays an important role in the cataracting-centrifuging transition, although most related works in the literature do not consider this. The particle shape also significantly affects the cataracting-centrifuging transition behavior. Nonspherical particles required lower rotation speeds than spherical particles to reach the centrifuging condition. The particle shape was shown to be related to the critical solid fraction (α_sc) from Schaeffer’s model, although further investigations are required to completely correlate particle spherecity with critical solid volume fractions.     

On the calculation of wall temperatures in the post dryout heat transfer region

A non-equilibrium post dryout heat transfer model for calculating the wall temperature distribution in vertical upflows is presented in this study. The model is based upon the three path heat transfer formulation developed by MIT researchers (Laverty & Rohsenow 1964, Forslund & Rohsenow 1968, Hynek et al. 1969 and Plummer et al. 1974) that involves heat transfer from wall to vapor, from wall to droplets in contact with the wall and from vapor to liquid droplets in the vapor core. Downstream gradients for the bulk vapor temperature, vapor quality, droplet size and vapor velocities are identical to those used by Hynek et al. (1969) and Plummer et al. (1974). Conditions at the dryout location are calculated using a modified version of a technique developed by Hynek et al. (1969).A procedure for determining an average droplet diameter based on a size distribution is introduced. Migration of droplets through the boundary layer and droplet deposition flux are predicted with the model of Gani? & Rohsenow (1979). Heat transfer from the wall to the impinging liquid droplets is calculated with a correlation by Holman & McGinnis (1969). Mechanisms contributing to wall to droplet heat transfer are identified as (a) droplet-wall contact, (b) intensive droplet evaporation inside the boundary layer, and (c) destruction of the boundary layer due to droplet migration to, and rebound from, the hot surface. The significance of the average droplet size and size distribution is demonstrated through its control over the free stream evaporation and droplet deposition rates.Predicted uniform heat flux wall temperature profiles for water, nitrogen and freon 12 are in good agreement with the data of Era et al. (1966), Bennett et al. (1967), Forslund & Rohsenow (1968), Ling et al. (1971), Groeneveld (1972) and Janssen & Kervinen (1975).     

Simulation du front d'écoulement dans les procédés de moulage des composites liquides

This work proposes a procedure of finite difference discretized in a system of curvilinear coordinates adapted to the shape of the saturated zone, to simulate a flow in the LCM (Liquid Composites Molding) process. Formulation and the numerical application of the procedure are described. We describe two configurations of injections. A good agreement is found between numerical, analytical and literature experimental results. To cite this article: M. Hattabi et al., C. R. Mecanique 333 (2005).     

Direct numerical simulation of fluid flow in a 5x5 square rod bundle

Characterization of parallel flow through rod bundles is of key importance in assessing the performance and safety of several engineering systems, including a majority of nuclear reactor concepts. Inhomogeneities in the bundle cross-section can present complex flow phenomena, including varying local conditions of turbulence. With the ever-increasing capabilities of high-performance computing, Direct Numerical Simulation (DNS) of turbulent flows is becoming more feasible. Through resolving all scales of turbulence, DNS can serve as a “numerical experiment,” and can provide substantial insight into flow physics, but at considerable computational cost. Thus to date, the DNS in open literature for rod bundle flows is relatively scarce, and largely limited to unit-cell domains. Since wall effects are important in rod bundle flows, a multiple-pin DNS study can expand understanding of rod bundle flows while providing valuable reference data for evaluating reduced-resolution techniques. In this work, DNS of a 5x5 square bare rod bundle representative of typical light water reactor fuel dimensions was performed using the spectral element code Nek5000. Turbulent microscales based on an advanced Reynolds-Averaged Navier–Stokes model were used to establish the required DNS resolution. Velocity and Reynolds stress fields are analyzed in detail, and invariant analysis is used for further investigation into flow physics. The results show stark changes in the structure of turbulence in the edge gaps, suggesting the presence of gap vortices in these regions. In addition, turbulent kinetic energy budgets are presented to more fully illustrate the various turbulent processes. These data can prove useful for rigorous evaluation of lower-fidelity turbulence modeling approaches.     

Couplage dans les alliages à mémoire de formeCoupling in shape memory alloys.

This Note concerns the study of the micromechanic behavior of shape memory alloys. The advantage of this model permits the coupling between the martensitic transformation and microstructural evolution observed after cycling. The model makes it possible to obtain consecutives equations, which explain at the same time, mechanical properties and the changing structures during the transformation. It provides original physical results on the global behavior of shape memory alloys. To cite this article: A. Alhamany et al., C. R. Mecanique 332 (2004).     

A damping term based on material properties for the volume-based contact dynamics model

The Gonthier et al. volume-based contact dynamics model addressed many different phenomena that influence the force of contact between two objects. This work extends Gonthier et al.'s work by proposing an alternate damping model based on material properties. The normal contact force based on volumetric interference information is derived using principles of mechanics of materials. The volume of interference and its rate of change are shown to be analogues to the material deformation and deformation rate. Simulations of the free direct central impact between two identical spheres are run using the proposed model, the Hunt–Crossley model and the Gonthier et al. model; these are compared to experimental data from the literature. This is followed by a discussion of those models.     

An asymptotic non-linear model for thin-walled rodsUn modèle asymptotique non-linéaire de poutres à parois minces

In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions. To cite this article: L. Grillet et al., C. R. Mecanique 332 (2004).     

Development of fracture facets from a crack loaded in mode I+III: Solution and application of a model 2D problem

It is experimentally well-known that a crack loaded in mode I+III propagates through formation of discrete fracture facets inclined at a certain tilt angle on the original crack plane, depending on the ratio of the mode III to mode I initial stress intensity factors. Pollard et al. (1982) have proposed to calculate this angle by considering the tractions on all possible future infinitesimal facets and assuming shear tractions to be zero on that which will actually develop. In this paper we consider the opposite case of well-developed facets; the stress field near the lateral fronts of such facets becomes independent of the initial crack and essentially 2D in a plane perpendicular to the main direction of crack propagation.To determine this stress field, we solve the model 2D problem of an infinite plate containing an infinite periodic array of cracks inclined at some angle on a straight line, and loaded through uniform stresses at infinity. This is done first analytically, for small values of this angle, by combining Muskhelishvili's (1953) formalism and a first-order perturbation procedure. The formulae found for the 2D stress intensity factors are then extended in an approximate way to larger angles by using another reference solution, and finally assessed through comparison with some finite element results.To finally illustrate the possible future application of these formulae to the prediction of the stationary tilt angle, we introduce the tentative assumption that the 2D mode II stress intensity factor is zero on the lateral fronts of the facets. An approximate formula providing the tilt angle as a function of the ratio of the mode III to mode I stress intensity factors of the initial crack is deduced from there. This formula, which slightly depends on the type of loading imposed, predicts somewhat smaller angles than that of Pollard et al. (1982).     

Torsion of an elastic rod whose cross-section is a parallelogram,trapezoid, or triangle,or has an arbitrary shape by the method of transformation to a rectangular domain

A coordinate transformation is used to take the domain of the rod cross-section to a rectangular domain for which the spectra of eigenfunctions and eigenvalues are known. The torsion function is represented as a generalized Fourier series to reduce the problem to solving a closed linear system of algebraic equations for the expansion coefficients. It is shown that these Fourier series converge absolutely, because the expansion coefficients decrease by a cubic law depending on the term number. We prove that the approximate solution in the form of a finite sum of the Fourier series converges to the exact solution. This theorem is generalized to the case of a rod cross-section of arbitrary shape.     

Development of a Low-Reynolds-number k-ω Model for FENE-P Fluids

A low-Reynolds-number k-ω model for Newtonian fluids has been developed to predict drag reduction of viscoelastic fluids described by the FENE-P model. The model is an extension to viscoelastic fluids of the model for Newtonian fluids developed by Bredberg et al. (Int J Heat Fluid Flow 23:731–743, 2002). The performance of the model was assessed using results from direct numerical simulations for fully developed turbulent channel flow of FENE-P fluids. It should only be used for drag reductions of up to 50 % (low and intermediate drag reductions), because of the limiting assumption of turbulence isotropy leading to an under-prediction of k, but compares favourably with results from k-ε models in the literature based on turbulence isotropy.     

Theoretical analysis of the channel die compression test—I. General considerations and finite deformation of F.C.C. crystals in stable lattice orientations

An extensive theoretical investigation of f.c.c. crystals under [110] loading in the channel die compression test is presented. Two lattice orientations known from experiment to be stable relative to the channel axes, through large deformations, are investigated for each of four hardening laws. These are Taylor's classical isotropic hardening rule, a 2-parameter empirical rule from the metallurgical literature, the “simple theory” of anisotropic latent hardening(Havner and Shalaby, Proc. R. Soc. A 358,47 (1977)), and a modification of the simple theory proposed by pfirce et al., Acta Met.30, 1087 (1982). Predictions of active systems, equal multiple-slip and consequent lattice stability, finite shape change, and lateral constraint stress are the same for all theories, corresponding to minimum rate of plastic work, and are in general agreement with experiments on copper crystals by Wonsiewicz and Chin, Met. Trans.1, 2715 (1970) and Wonsiewiczet al., Met. Trans.2, 2093 (1971). The predictions of latent hardening differ among the theories, however, depending upon whether there is relative rotation of material and lattice. The potential significance of experimental studies of latent hardening in these particular stable lattice orientations is emphasized.     

Further consideration of the shape-factor relationship for turbulent boundary layers

The lag-entrainment predictive scheme developed by Green et al. has been modified to include the pressure-gradient parameter Π₁. In the original model suggested by Green et al. the mass-flow shape factor H₁ is related to the common shape factor H, H₁ = f(H). In the present model H₁ is related to H, Reynolds number based on the local momentum thickness θ, and Π₁; thus H₁ = f(H, R_eθ, Π₁). The modified formula for H₁, is introduced into the original lag-entrainment integral model. Calculations are made to examine the present model for the predictions of the development of boundary layers approaching separation studied experimentally by the authors. Slightly improved predictions are obtained using the model developed by El Telbany et al. However, the present model proved to give an improved representation of the development of wall shear stress in cases the two-equation turbulence model proved to be unsuccessful.     

On the optimal shape of compressed rotating rod with shear and extensibility

In this paper, the optimal shape of a compressed rotating rod which maintains stability against buckling is presented. In the rod modeling, extensibility along the rod axis and shear stress is taken into account. Using Pontryagin's maximum principle, the optimization problem is formulated with a fourth order boundary value problem. The optimally shaped compressed rotating (fixed-free) rod has a finite cross-sectional area on the free end. This shape is qualitatively different from that suggested by the Bernoulli-Euler theory with zero cross-sectional area on the free end. In addition, the Bernoulli-Euler theory overestimates the buckling load, and this effect is more significant in the optimally shaped rod than for the corresponding constant cross-sectional rod consisting of the same material volume and length. In order to show this effect, it is necessary to use a generalized constitutive model which takes real material properties, such as axial extensibility and shear stress into account. Particularly, the solution of this generalized problem, obtained for thin rods, approaches the classical solution predicted by the Bernoulli-Euler theory.     

A strength reliability model by Markov process of unidirectional composites with fibers placed in hexagonal arrays

This paper proposes a strength reliability model based on a Markov process for unidirectional composites with fibers in a hexagonal array. The model assumes that a group of fiber breaking points, a so-called cluster, evolves with increased stress. The cluster evolution process branches because of various fiber-breakage paths. Load-sharing structure of intact fibers around clusters was estimated from geometric and mechanical local load-sharing rules. Composites fracture if a cluster achieves a critical size, so the model expresses a fracture criterion by setting an absorbing state. Next, the author constituted a state transition diagram concerning cluster evolutions of 1-fiber to 7-fiber breaks and analytically solved simultaneous differential equations obtained from the diagram. Results showed that, as critical cluster size increases, slope of the fracture probability distribution is given in a Weibull probability scale as follows: m_c=i×m_f (i, the number of broken fibers in a cluster; m_c and m_f, Weibull shape parameters for fracture probabilities of a critical cluster and fiber strength, respectively). This relation between m_c and m_f had been shown by Smith et al. [Proc. R. Soc. London, A 388 (1983) 353–391], but the present study demonstrated it analytically without any lower tail of the Weibull distribution used in that paper. In addition, the present model can be approximated by a one-state birth model.     

THE SERIAL MODEL IN LIQUID FLUIDIZATION AND ITS LIMITATIONS

The serial model of Epstein et al. (1981) is demonstrated to predict successfully the overall voidage of the mixed layer of incompletely segregated binary and ternary mixtures of liquid-fluidized solids that differ only in shape. Some speculations are then offered on the conditions for the success and failure of this model. Finally, methods proposed in the literature for estimating the deviations from the serial model for a binary-solids bed, which manifest themselves as a bed contraction, are discussed, with emphasis on packing models.     

On the stability problem of Nicolai with variable cross-section and compressibility effect

We consider the problem of Nicolai on dynamic stability of an elastic cantilever rod loaded by an axial compressive force and tangential twisting torque in continuous formulation. The rod is assumed to be non-uniform, i.e. having variable cross-section with non-equal principal moments of inertia. New linear equations and boundary conditions are derived from nonlinear governing equations. These equations form the basis for analytical and numerical studies. The important new details of this formulation include the pre-twisting effect due to the torque and compressibility of the rod. General formulae for the influence of small geometrical imperfections to the stability region are derived and numerical examples are presented.