Collision operator including large-angle scattering for moderately coupled plasma

A new approach has been developed to treat the large-angle as well as the small-angle binary collisions in high temperature and high density plasmas when the test particle distribution function f_α is even function about the test velocity and the relations of the mass and the velocity between the test particles and the field particles are satisfied with m_αm_β (such as electron–ion collision or Lorentz-gas model) and . With the approach, the Boltzmann collision operator is derived to be suitable for the plasma considered as weakly coupled (Coulomb logarithm ) and moderately coupled , i.e., for the electron–ion coupling constant Γ_ei<0.1. The modified collision operator has a direct and practical connection to the Rosenbluth potentials, the new reduced electron–ion collision operator differs from the original Fokker–Planck operator for Coulomb collisions by terms of order . Moreover, some calculations of relaxation rate and transport properties are given for new reduced electron–ion collision operator that shows corrections.     

Explicit conditions for the existence of exceptional body waves and subsonic surface waves in anisotropic elastic solids

It is known that a subsonic surface (Rayleigh) wave exists in an anisotropic elastic half-space x₂  0 if the first transonic state is not of Type 1. If the first transonic state is of Type 1 but the limiting wave is not exceptional, a subsonic surface wave exists. If the first transonic state is of Type 1 and the limiting wave is exceptional, a subsonic surface wave exists when . It is shown that an exceptional body wave is necessarily an exceptional transonic wave, and could be an exceptional limiting wave. Only two wave speeds are possible for an exceptional body wave. We present explicit conditions in terms of the reduced elastic compliances for the existence of an exceptional body wave. If an exceptional body wave exists, conditions are given for identifying whether the transonic state is of Type 1. Hence, through the existence of an exceptional body wave we provide explicit conditions for the existence of a subsonic surface wave with the exception when needs to be computed.     

Regularity criteria for the 3D MHD equations in terms of the pressure

In this paper we consider the regularity criteria for weak solutions to the 3D MHD equations. It is proved that under the condition b being in the Serrin's regularity class, if the pressure p belongs to L^α,γ with or the gradient field of pressure p belongs to L^α,γ with on [0,T], then the solution remains smooth on [0,T].     

Integration approach of the Couette inverse problem of powder type self-compacting concrete in a wide-gap concentric cylinder rheometer

For powder type self-compacting concrete (SCC) mixes, commonly used in Belgium, a shear thickening (Herschel–Bulkley) flow behaviour of the fresh mixes is quite often observed.A longstanding problem in rheometry is the so-called “Couette inverse problem”, where one tries to derive the flow curve from the torque measurements T(N) in a (wide-gap) concentric cylinder (Couette) rheometer, with T the torque registered at the inner, stationary cylinder and N the rotational velocity of the outer, rotating, cylinder.In this paper, the Couette inverse problem is approached by means of the integration method in order to convert T(N) into for a wide-gap (R_o/R_i = 1.45) concentric cylinder rheometer. The approach consists in the decoupling of the flow resistance and the power-law flow behaviour after exceeding the flow resistance. The integration approach is validated by experimental verification with different powder type SCC mixtures. By means of illustration, the results of one limestone powder type SCC mixture with different superplasticizer contents are shown in this paper.     

Computing Green's function of elasticity in a half-plane with impedance boundary condition

This Note presents an effective and accurate method for numerical calculation of the Green's function G associated with the time harmonic elasticity system in a half-plane, where an impedance boundary condition is considered. The need to compute this function arises when studying wave propagation in underground mining and seismological engineering. To theoretically obtain this Green's function, we have drawn our inspiration from the paper by Durán et al. (2005), where the Green's function for the Helmholtz equation has been computed. The method consists in applying a partial Fourier transform, which allows an explicit calculation of the so-called spectral Green's function. In order to compute its inverse Fourier transform, we separate as a sum of two terms. The first is associated with the whole plane, whereas the second takes into account the half-plane and the boundary conditions. The first term corresponds to the Green's function of the well known time-harmonic elasticity system in (cf. J. Dompierre, Thesis). The second term is separated as a sum of three terms, where two of them contain singularities in the spectral variable (pseudo-poles and poles) and the other is regular and decreasing at infinity. The inverse Fourier transform of the singular terms are analytically computed, whereas the regular one is numerically obtained via an FFT algorithm. We present a numerical result. Moreover, we show that, under some conditions, a fourth additional slowness appears and which could produce a new surface wave. To cite this article: M. Durán et al., C. R. Mecanique 334 (2006).     

Analysing flow and heat transfer of a viscoelastic fluid over a semi-infinite horizontal moving flat plate

This paper presents a numerical study of the flow and heat transfer of an incompressible homogeneous second grade type fluid above a flat plate moving with constant velocity U. Such a viscoelastic fluid is at rest and the motion is created by the sheet. The effects of the non-Newtonian nature of the fluid are governed by the local Deborah number K (the ratio between the relaxation time of the fluid and the characteristic time of the flow). When , a new analytical solution for this flow is presented and the effects of fluid's elasticity on flow characteristics, dimensionless stream function and its derivatives are analysed in a wide domain of K. A novel result of the analysis is that a change in the flow solution's behaviour occurs when the dimensionless stream function at the edge of the boundary layer, f_∞, equals 1.0. It is found that velocity at a point decreases with increase in the elasticity of the fluid and, as expected, the amount of fluid entrained diminishes when the effects of fluid's elasticity are augmented. In our heat transfer analyses we assume that the surface temperature has a power-law variation. Two cases are studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the sheet with prescribed heat flux (PHF case). Local similarity heat-transfer solutions are given for PST case when s=2 (the wall temperature parameter) whereas when a similarity solution takes place in the case of prescribed wall heat flux. The numerical results obtained are fairly in good agreement with the aforementioned analytical ones.     

Multiscale simulation of viscoelastic free surface flows

A micro–macro approach based on combining the Brownian configuration fields (BCF) method [M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] with an Arbitrary Lagrangian–Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time-dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on the conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409]. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers, (where λ is the characteristic relaxation time of polymer, and is the characteristic rate of strain) compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field.     

Fracture strength for a high strength steel bridge cable wire with a surface crack

In this paper, the fracture strength of a cracked suspension bridge wire is determined based on linear elastic fracture mechanics (LEFM). The wire is 5 mm in diameter, with an original ultimate strength of 1725 MPa and ultimate elongation that ranges between 5.5% and 6%. The average value of for the wire fracture toughness, K_C, was recently evaluated by the author. The state of practice is to use the ultimate strength of the cracked wire as obtained from tensile tests. This approach may overestimate the strength of the wire due to possible delamination and crack tip plasticity. A case study for a group of in situ wire breaks retrieved from a suspension bridge cable is presented. The failure analysis is performed based on both the fracture toughness criterion and the net section theory. The fracture toughness criterion produced more realistic results for the fracture strength of the wire. The decline in the fracture toughness and the corresponding reduction in the fracture strength of cracked degraded wire are predicted making use of the strain energy density criterion.     

Bifurcation analysis of an arch structure with parametric and forced excitation

The subharmonic bifurcation and universal unfolding problems are discussed for an arch structure with parametric and forced excitation in this paper. The amplitude–frequency curve and some dynamical behavior have been shown for this class of problems by Liu et al. Here, by means of singularity theory, in the case of strict 1:2 internal resonance, the bifurcation behavior of the amplitude with respect to a parameter (which is related to the amplitude of the live load imposed on the arch structures) is studied. The results indicate that it is a high codimensional bifurcation problem with codimension 5, and the universal unfolding is given. From the mechanical background, 20 forms of two parameter unfoldings with some constraints are studied. The transition sets in the parameter plane and the bifurcation diagrams are plotted. The results obtained in this paper present some new dynamic buckling patterns and abundant bifurcation phenomena.     

Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist, and held at both endpoints at infinity. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically nonlinear. The problem is formulated as a nonlinear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. We obtain a family of equilibrium solutions depending on a single loading parameter (proportional to applied twisting moment divided by square root of pulling force), which are asymptotically valid in the limit of a loose knot, ε→0. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium.     

Analysis of regular and irregular acoustic streaming patterns in a rectangular enclosure

This study reports an experimental investigation of the non-linear phenomena of regular (classical) and irregular streaming patterns generated in an air-filled rigid-walled square channel subjected to the acoustic standing waves of different frequencies and intensities. The interaction of acoustic waves and thermoviscous fluids is responsible for these phenomena. The resonator’s walls are maintained at isothermal condition. Synchronized particle image velocimetry (PIV) technique has been used to measure the streaming velocity fields. The experimental results show that at a given excitation frequency, regular streaming flow patterns are observed up to a certain value of the excitation amplitude. As the amplitude increases beyond this limit, the regular streaming is distorted to an irregular flow structure. The regular and irregular streaming are classified in terms of streaming Reynolds number . It is found that for Re_s2<50, classical streaming flow patterns are established and then deform to irregular and complex shapes as Re_s2 exceeds 50.     

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

Uniformly valid solution of limit cycle of the Duffing–van der Pol equation

The limit cycle of the Duffing–van der Pol equation is studied. By considering the product of the frequency ω of the limit cycle and the coefficient ε as an independent parameter μ=εω, an equivalent equation is obtained and then solved by Liao’s homotopy analysis method. The frequency ω is deduced as a function of μ and δ. This function provides us with an algebraic equation for ω, according to which we have an analytical approximation for the frequency. Numerical examples show that the attained approximation is very accurate. More importantly, the results are uniformly valid for all positive values of ε.     

Damage evolution of ductile material under impact loading

Nucleation, growth and coalescence of micro-voids result in the fracture of materials. Most mathematical models neglect nucleation and introduce initial damage, assuming it as a material constant. However, the original damage, which is formed during material working, is a material constant. The initial damage is a model parameter and depends on the load. Apparently, the predictability of such a model is poor.This paper made comparison and analysis of the four classical void growth models and showed their similarities. At the beginning of damage evolution, all the models follow a linear relationship in the form , where c is the size of micro voids and k is a parameter which relates the material and loading condition. With the concept of statistical micro-damage and the assumption of uniform void radius for new voids, a damage evolution equation was deduced based on the above void growth model. With this equation the effects of nucleation and growth at the beginning of the damage stage on the whole process of damage evolution can be calculated. The transition time from the nucleation dominant phase to the growth dominant phase can be determined. When the transition time is applied to the damage failure model of ductile material proposed by Johnson, the initial damage (f₀), a model parameter in the original model, can also be determined. The results of the derived damage evolution equation agree well with the previous research results.     

Processing induced size effects in plastic yielding upon miniaturisation

Size effects in metals have received considerable attention in literature in the last decades. For preparing specimens dedicated processing techniques, such as laser-cutting, micro-milling, turning, etc., are used. Most of these processing methods intrinsically damage crystals just below the worked surface. In macroscopic applications, the effect on the overall mechanical behaviour can safely be neglected in most cases. Upon miniaturisation, however, the influence of the affected region becomes more important and may induce a processing induced size effect, which is far from negligible. Processing induced size effects are analysed by carefully characterising the plastic yielding in uniaxial tension of rectangular, -thick aluminium sheet specimens, with a well-defined homogeneous microstructure containing through-thickness grains. The specimens are processed to different widths by three independent machining techniques: (1) laser-cutting, (2) mechanical cutting, and (3) extensive grinding from a larger width. These independent techniques all result in a distinct processing induced size effect upon miniaturisation, i.e. an increase of up to 200% in yield stress for a decrease from about 12 to 3 grains over the specimen width. Using a simple Taylor averaging model, it is shown that the yield stress in the affected edge region increased to 210–350% of its initial (or bulk) value. In addition, it is found that even a prolonged anneal near the melting temperature can only partially remove the processing induced size effect. The results clearly demonstrate that processing induced size effects have to be considered in the design of miniaturised devices and parts as well as in scientific research relying on the testing of manufactured small-scale test specimens.     

Crack repair using an elastic filler

The effect of repairing a crack in an elastic body using an elastic filler is examined in terms of the stress intensity levels generated at the crack tip. The effect of the filler is to change the stress field singularity from order 1/r^1/2 to 1/r^(1-λ) where r is the distance from the crack tip, and λ is the solution to a simple transcendental equation. The singularity power (1-λ) varies from (the unfilled crack limit) to 1 (the fully repaired crack), depending primarily on the scaled shear modulus ratio γ_r defined by G₂/G₁=γ_rε, where 2πε is the (small) crack angle, and the indices (1, 2) refer to base and filler material properties, respectively. The fully repaired limit is effectively reached for γ_r≈10, so that fillers with surprisingly small shear modulus ratios can be effectively used to repair cracks. This fits in with observations in the mining industry, where materials with G₂/G₁ of the order of 10^-3 have been found to be effective for stabilizing the walls of tunnels. The results are also relevant for the repair of cracks in thin elastic sheets.     

Effective flow of a viscous liquid through a helical pipe

We study the flow of a viscous fluid through a pipe with helical shape parameterized with , where the small parameter stands for the distance between two coils of the helix. The pipe has small cross-section of size . Using the asymptotic analysis of the microscopic flow described by the Navier–Stokes system, with respect to the small parameter that tends to zero, we find the effective fluid flow described by an explicit formula of the Poisseuile type including a small distorsion due to the particular geometry of the pipe. To cite this article: E. Marušić-Paloka, I. Pažanin, C. R. Mecanique 332 (2004).

Résumé

On considère un écoulement dans un tube de section circulaire et de forme hélicoïdale paramétré par , où est la distance entre deux tours de la spirale. Le rayon de la section du tube est lui aussi supposé égal à . A partir de l'écoulement microscopique décrit par le système de Navier–Stokes et en utilisant l'analyse asymptotique par rapport à ce petit paramètre on obtient l'écoulemment effectif décrit par une formule explicite de type Poiseuille associée à une petite déviation due à la géometrie du tube. Pour citer cet article : E. Marušić-Paloka, I. Pažanin, C. R. Mecanique 332 (2004).     

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.     

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.