Geometrical properties of the vorticity vector derived using large-eddy simulation

In this paper, the geometrical properties of the resolved vorticity vector derived from large-eddy simulation are investigated using a statistical method. Numerical tests have been performed based on a turbulent Couette channel flow using three different dynamic linear and nonlinear subgrid-scale stress models. The geometrical properties of have a significant impact on various physical quantities and processes of the flow. To demonstrate, we examined helicity and helical structure, the attitude of with respect to the eigenframes of the resolved strain rate tensor and negative subgrid-scale stress tensor -τ_ij, enstrophy generation, and local vortex stretching and compression. It is observed that the presence of the wall has a strong anisotropic influence on the alignment patterns between and the eigenvectors of , and between and the resolved vortex stretching vector. Some interesting wall-limiting geometrical alignment patterns and probability density distributions in the form of Dirac delta functions associated with these alignment patterns are reported. To quantify the subgrid-scale modelling effects, the attitude of with respect to the eigenframe of -τ_ij is studied, and the geometrical alignment between and the Euler axis is also investigated. The Euler axis and angle for describing the relative rotation between the eigenframes of -τ_ij and are natural invariants of the rotation matrix, and are found to be effective for characterizing a subgrid-scale stress model and for quantifying the associated subgrid-scale modelling effects on the geometrical properties of .     

Limit analysis of multi-layered plates. Part I: The homogenized Love-Kirchhoff model

The purpose of this paper is to determine , the overall homogenized Love-Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate, and to study the relationship between the 3D and the homogenized Love-Kirchhoff plate limit analysis problems. In the Love-Kirchhoff model, the generalized stresses are the in-plane (membrane) and the out-of-plane (flexural) stress field resultants. The homogenization method proposed by Bourgeois [1997. Modélisation numérique des panneaux structuraux légers. Ph.D. Thesis, University Aix-Marseille] and Sab [2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. C. R. Méc. 331, 641-646] for in-plane periodic rigid perfectly plastic plates is justified using the asymptotic expansion method. For laminated plates, an explicit parametric representation of the yield surface is given thanks to the π-function (the plastic dissipation power density function) that describes the local strength domain at each point of the plate. This representation also provides a localization method for the determination of the 3D stress components corresponding to every generalized stress belonging to . For a laminated plate described with a yield function of the form , where σ^u is a positive even function of the out-of-plane coordinate x₃ and is a convex function of the local stress σ, two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials ( in the skins and in the core) is studied. It is found that, for small enough contrast ratios (), the normalized strength domain is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticité. Eyrolles, Paris].     

Discrete dislocation plasticity analysis of the wedge indentation of films

The plane strain indentation of single crystal films on a rigid substrate by a rigid wedge indenter is analyzed using discrete dislocation plasticity. The crystals have three slip systems at ±35.3^° and 90^° with respect to the indentation direction. The analyses are carried out for three values of the film thickness, 2, 10 and , and with the dislocations all of edge character modeled as line singularities in a linear elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated through a set of constitutive rules. Over the range of indentation depths considered, the indentation pressure for the 10 and thick films decreases with increasing contact size and attains a contact size-independent value for contact lengths . On the other hand, for the films, the indentation pressure first decreases with increasing contact size and subsequently increases as the plastic zone reaches the rigid substrate. For the 10 and thick films sink-in occurs around the indenter, while pile-up occurs in the film when the plastic zone reaches the substrate. Comparisons are made with predictions obtained from other formulations: (i) the contact size-independent indentation pressure is compared with that given by continuum crystal plasticity; (ii) the scaling of the indentation pressure with indentation depth is compared with the relation proposed by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 43, 411-423]; and (iii) the computed contact area is compared with that obtained from the estimation procedure of Oliver and Pharr [1992. An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7, 1564-1583].     

Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension

The membrane deflection experiment developed by Espinosa and co-workers was used to examine size effects on mechanical properties of free-standing polycrystalline FCC thin films. We present stress-strain curves obtained on films 0.2, 0.3, 0.5 and thick including specimen widths of 2.5, 5.0, 10.0 and for each thickness. Elastic modulus was consistently measured in the range of 53- for Au, 125- for Cu and 65- for Al. Several size effects were observed including yield stress variations with membrane width and film thickness in pure tension. The yield stress of the membranes was found to increase as membrane width and thickness decreased. It was also observed that thickness plays a major role in deformation behavior and fracture of polycrystalline FCC metals. A strengthening size scale of one over film thickness was identified. In the case of Au free-standing films, a major transition in the material inelastic response occurs when thickness is changed from 1 to . In this transition, the yield stress more than doubled when film thickness was decreased, with the thick specimen exhibiting a more brittle-like failure and the thick specimen exhibiting a strain softening behavior. Similar plasticity size effects were observed in Cu and Al. Scanning electron microscopy performed on Au films revealed that the number of grains through the thickness essentially halved, from approximately 5 to 2, as thickness decreased. It is postulated that this feature affects the number of dislocations sources, active slip systems, and dislocation motion paths leading to the observed strengthening. This statistical effect is corroborated by the stress-strain data in the sense that data scatter increases with increase in thickness, i.e., plasticity activity.The size effects here reported are the first of their kind in the sense that the measurements were performed on free-standing polycrystalline FCC thin films subjected to macroscopic homogeneous axial deformation, i.e., in the absence of deformation gradients, in contrast to nanoindentation, beam deflection, and torsion, where deformation gradients occur. To the best of our understanding, continuum plasticity models in their current form cannot capture the observed size scale effects.     

Theoretical and experimental analysis of longitudinal wave propagation in cylindrical viscoelastic rods

Wave propagation in viscoelastic rods is encountered in many applications including studies of impact and fracture under high strain rates and characterization of the dynamic behavior of viscoelastic materials. For viscoelastic materials, both material and geometric dispersion are possible when the diameter of the rod is of the same order as the wavelength. In this work, we simplify the Pochhammer frequency equation for low and intermediate loss viscoelastic materials and formulate corrections for geometric dispersion for both the phase velocity and attenuation. The formulation is then experimentally verified with measurements of the phase velocity and attenuation in commercial polymethylmethacrylate rods that are 12 and in diameter. Without correcting for geometric dispersion, the usable frequency range for determining the phase velocity and attenuation for the rod is about , and about for the rod. Using the correction procedure developed here, it was possible to accurately determine the phase velocity and attenuation up to frequencies exceeding for the rod and for the rod. These corrections are applicable to many polymers and other viscoelastic materials. From thereon, the viscoelastic properties of the material can be determined over a wide range of frequencies.     

Deformation of FCC nanowires by twinning and slip

We present atomistic simulations of the tensile and compressive loading of single crystal face-centered cubic (FCC) nanowires with and orientations to study the propensity of the nanowires to deform via twinning or slip. By studying the deformation characteristics of three FCC materials with disparate stacking fault energies (gold, copper and nickel), we find that the deformation mechanisms in the nanowires are a function of the intrinsic material properties, applied stress state, axial crystallographic orientation and exposed transverse surfaces. The key finding of this work is the first order effect that side surface orientation has on the operant mode of inelastic deformation in both and nanowires. Comparisons to expected deformation modes, as calculated using crystallographic Schmid factors for tension and compression, are provided to illustrate how transverse surface orientations can directly alter the deformation mechanisms in materials with nanometer scale dimensions.     

A perturbation-incremental method for strongly non-linear non-autonomous oscillators

A perturbation-incremental method is extended for the analysis of strongly non-linear non-autonomous oscillators of the form , where g(x) and are arbitrary non-linear functions of their arguments, and ε can take arbitrary values. Limit cycles of the oscillators can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at period-doubling bifurcation along a frequency-response curve is made simple by the present method. Subsequent continuation of an emanating branch is also discussed.     

Micromechanics of residual stress and texture development due to poling in polycrystalline ferroelectric ceramics

This paper presents micromechanics based analysis of elastic strain and changes in the texture of poled polycrystalline ferroelectric PZT ceramics for direct comparison with synchrotron X-ray measurements. The grains are modelled as spherical inclusions, to which transformation strains are assigned depending on the fractions of different ferroelectric domains. Eshelby's inclusion problem with the classical self-consistent method is applied to evaluate the elastic state of the grains. In particular, the elongation due to lattice elastic strain is calculated as a function of inclination Ψ relative to the polar axis. The ratio of diffraction peak intensities, corresponding to the domain fractions, is also expressed as a function of Ψ. This analysis identifies the special character of the reflection, for which the lattice strain along in the stress free state is independent of ferroelectric domain population and hence unaffected by poling. The elongation due to the lattice strain parallel to and peak intensity ratio are expressed in terms of the overall macroscopic strain of a poled specimen, each having a dependence.     

Glass-modified stress waves for adhesion measurement of ultra thin films for device applications

Laser-generated stress wave profiles with rarefaction shocks (almost zero post-peak decay times) have been uncovered in different types of glasses and presented in this communication. The rise time of the pulses was found to increase with their amplitude, with values reaching as high as . This is in contrast to measurements in other brittle crystalline solids where pulses with rise times of and post-peak decay times of were recorded. The formation of rarefaction shock is attributed to the increased compressibility of glasses with increasing pressures. This was demonstrated using a one-dimensional nonlinear elastic wave propagation model in which the wave speed was taken as a function of particle velocity. The technological importance of these pulses in measuring the tensile strength of very thin film interfaces is demonstrated by using a previously developed laser spallation experiment in which a laser-generated compressive stress pulse in the substrate reflects into a tensile wave from the free surface of the film and pries off its interface at a threshold amplitude. Because of the rarefaction shock, glass-modified waves allow generation of substantially higher interfacial tensile stress amplitudes compared with those with finite post-peak decay profiles. Thus, for the first time, tensile strengths of very strong and ultra thin film interfaces can be measured. Results presented here indicate that interfaces of 185-nm-thick films, and with strengths as high as , can be measured. Thus, an important advance has been made that should allow material optimization of ultra thin layer systems that may form the basis of future MEMS-based microelectronic, mechanical and clinical devices.     

Plasticity of metal wires in torsion: Molecular dynamics and dislocation dynamics simulations

The orientation dependent plasticity in metal nanowires is investigated using molecular dynamics and dislocation dynamics simulations. Molecular dynamics simulations show that the orientation of single crystal metal wires controls the mechanisms of plastic deformation. For wires oriented along , dislocations nucleate along the axis of the wire, making the deformation homogeneous. These wires also maintain most of their strength after yield. In contrast, wires oriented along and directions deform through the formation of twist boundaries and tend not to recover when high angle twist boundaries are formed. The stability of the dislocation structures observed in molecular dynamics simulations are investigated using analytical and dislocation dynamics models.     

Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G=curlH^p, with H^p the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates , and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which

(A)

on a subsurface S_hard of the boundary

for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor , a result that motivates replacing (A) with the microhard condition

(B)

on the subsurface S_hard.

We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across S_hard.What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions.Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.     

A new analytical technique to find periodic solutions of non-linear systems

Based on the classical harmonic balance method a new technique is presented to determine higher approximate periodic solutions of the non-linear differential equations. The new method is systematic and simple. The solution covers the general initial value problem (i.e., for while the existing solution is determined for a particular case, especially for . The solution is easily transformed to perturbation solution. The method is used in various non-linear problems possessing second and more than second derivatives.