Uniaxial Modeling of Multivariant Shape-Memory Materials with Internal Sublooping using Dissipation Functions

Models for the macroscopic behavior of Shape Memory Materials can be conveniently constructed within the Ziegler–Green–Naghdi framework where all the constitutive information is encoded in two ingredients: the free energy and the dissipation function. In a previous work, we have proposed various expressions for the basic functions suitable to model pseudoelasticity with complete transformations cycles. In this work we consider additional effects due to Martensite reorientation and to transformation reversal prior to transformation completion. The new constitutive model allows for the modeling of a variety of effects including: shape memory associated with thermally induced transformation, internal pseudoelastic subloops and the determination of limit cycles associated with repetitive stress cycling.     

Torsional Swelling of a Hyperelastic Tube with Helically Wound Reinforcement

This paper examines the combination of radial deformation with torsion for a circular cylindrical tube composed of a transversely isotropic hyperelastic material subject to finite deformation swelling. The stored energy function involves separate matrix and fiber contributions such that the fiber contribution is minimized when the fiber direction is at a natural length. This natural length is not affected by the swelling. Hence swelling preferentially expands directions that are orthogonal to the fiber. The swelling itself is described via a swelling field that prescribes the local free volume at each location in the body. Such a treatment is a relatively simple generalization of the conventional incompressible theory. The direction of transverse isotropy associated with the fiber reinforcement is described by a helical winding about the tube axis. The swelling induced preferential expansion orthogonal to this direction induces the torsional aspect of the deformation. For a specific class of strain energy functions we find that the twist increases with swelling and approaches a limiting asymptotic value as the swelling becomes large. The fibers reorient such that fibers at the inner portion of the tube assume a more circumferential orientation whereas, at least for small and moderate swelling, the fibers in the outer portion of the tube assume a more axial orientation. For large swelling the fibers in the outer portion of the tube reorient beyond the axial orientation, and so are described by helices with orientation in the opposite sense to that in the reference configuration.      

Edge singularities and structure of the 3-D Williams expansion

The elastic solution in a vicinity of a re-entrant wedge can be described by a Williams like expansion in terms of powers of the distance to a point on the edge. This expansion has a particular structure due to the invariance of the problem by translation parallel to the edge. We show here that some terms, so-called primary solutions, derive directly from solutions to the 2-D corner problem posed in the orthogonal cross section of the domain. The others, baptized shadow functions, derive of the primary solutions by integration along the axis parallel to the edge. This 3-D Williams expansion is shown to be equivalent to the edge expansion proposed by Costabel et al. [M. Costabel, M. Dauge, Z. Yosibash, A quasidual function method for extracting edge stress intensity functions, SIAM J. Math. Anal. 35 (5) (2004) 1177–1202]. To cite this article: T. Apel et al., C. R. Mecanique 336 (2008).     

Anisotropic hardening and non-associated flow in proportional loading of sheet metals

Conventional isotropic hardening models constrain the shape of the yield function to remain fixed throughout plastic deformation. However, experiments show that hardening is only approximately isotropic under conditions of proportional loading, giving rise to systematic errors in calculation of stresses based on models that impose the constraint. Five different material data for aluminum and stainless steel alloys are used to calibrate and evaluate five material models, ranging in complexity from a von Mises’ model based on isotropic hardening to a non- associated flow rule (AFR) model based on anisotropic hardening. A new model is described in which four stress–strain functions are explicitly integrated into the yield criterion in closed form definition of the yield condition. The model is based on a non-AFR so that this integration does not affect the accuracy of the plastic strain components defined by the gradient of a separate plastic potential function. The model not only enables the elimination of systematic errors for loading along the four loading conditions, but also leads to a significant reduction of systematic errors in other loading conditions to no higher than 1.5% of the magnitude of the predicted stresses, far less that errors obtained under isotropic hardening, and at a level comparable to experimental uncertainty in the stress measurement. The model is expected to lead to a significant improvement in stress prediction under conditions dominated by proportional loading, and this is expected to directly improve the accuracy of springback, tearing, and earing predictions for these processes. In addition, it is shown that there is no consequence on MK necking localization due to the saturation of the yield surface in pure shear that occurs with the aluminum alloys using the present model.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

Use of Faraday instabilities to enhance fuel pulverisation in air-blast atomisers

The atomization of liquids into a spray is an important process in many industrial applications and particularly in the aero-engine sector. Conventional air-blast injectors in aircraft engines today use aerodynamic shearing effects to atomize the liquid fuel. However, at operating conditions where the air velocity is below 30 m/s (such as ground start and high altitude restart) the atomization quality is poor. Consequently combustion is less efficient with high pollutant emissions. The objective of this study is to validate a new concept of injector which couples the shearing effects with the principle of ultrasonic atomization. The latter consists of using piezoelectric actuators to generate the oscillations of a wall in contact with the liquid film. This excitation perpendicular to the liquid film surface creates Faraday instabilities at the liquid/air interface. Amplitudes higher than a defined threshold value induce the break-up of ligaments and the formation of droplets. To cite this article: M. Boukra et al., C. R. Mecanique 337 (2009).     

Low-Prandtl-number Bénard–Marangoni convection in a vertical magnetic field

The effect of a homogeneous magnetic field on surface-tension-driven Bénard convection is studied by means of direct numerical simulations. The flow is computed in a rectangular domain with periodic horizontal boundary conditions and the free-slip condition on the bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. Two- and three-dimensional flows are computed for either vanishing or small Prandtl number, which are typical of liquid metals. The main focus of the paper is on a qualitative comparison of the flow states with the non-magnetic case, and on the effects associated with the possible near-cancellation of the nonlinear and pressure terms in the momentum equations for two-dimensional rolls. In the three-dimensional case, the transition from a stationary hexagonal pattern at the onset of convection to three-dimensional time-dependent convection is explored by a series of simulations at zero Prandtl number.     

Analysis and reconstruction of a pulsed jet in crossflow by multi-plane snapshot POD

In this work, snapshot proper orthogonal decomposition (POD) is used to study a pulsed jet in crossflow where the velocity fields are extracted from stereoscopic particle image velocimetry (SPIV) results. The studied pulsed jet is characterized by a frequency f = 1 Hz, a Reynolds number Re _j  = 500 (based on the mean jet velocity ${\overline{U}_{j}}$  = 1.67 cm/s and a mean velocity ratio of R = 1). Pulsed jet and continuous jet are compared via mean velocity field trajectory and Q criterion. POD results of instantaneous, phase-averaged and fluctuating velocity fields are presented and compared in this paper. Snapshot POD applied on one plane allows us to distinguish an organization of the first spatial eigenmodes. A distinction between “natural modes” and “pulsed modes” is achieved with the results obtained by the pulsed and unforced jet. Secondly, the correlation tensor is established with four parallel planes (multi-plane snapshot POD) for the evaluation of volume spatial modes. These resulting modes are interpolated and the volume velocity field is reconstructed with a minimal number of modes for all the times of the pulsation period. These reconstructions are compared to orthogonal measurements to the transverse jet in order to validate the obtained three-dimensional velocity fields. Finally, this POD approach for the 3D flow field reconstruction from experimental data issued from planes parallel to the flow seems capable to extract relevant information from a complex three-dimensional flow and can be an alternative to tomo-PIV for large volume of measurement.     

Experimental exploration of underexpanded supersonic jets

Two underexpanded free jets at fully expanded Mach numbers $M_\mathrm{j}$ = 1.15 and 1.50 are studied. Schlieren visualizations as well as measurements of static pressure, Pitot pressure and velocity are performed. All these experimental techniques are associated to obtain an accurate picture of the jet flow development. In particular, expansion, compression and neutral zones have been identified in each shock cell. Particle lag is considered by integrating the equation of motion for particles in a fluid flow and it is found that the laser Doppler velocimetry is suitable for investigating shock-containing jets. Even downstream of the normal shock arising in the $M_\mathrm{j}$ = 1.50 jet, the measured gradual velocity decrease is shown to be relevant.     

Diophantine Geometry of the Torsion of a Drinfeld Module

We prove an analogue of the Manin-Mumford conjecture for Drinfeld modules of generic characteristic.