The criteria for identifying the type of critical points

Summary It is well known that the buckling of a structure includes two kinds of different physical phenomena. One is called snap-through buckling and the other is called bifurcation. In this paper, the criteria to identify the two different unstable types are developed. The analysis in the present paper shows that the type of critical points can be directly determined by the incremental equilibrium equations in the general finite element analysis and that it only depends on whether or not the load increment does work through the unstable mode associated with the critical load.

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Kriterien zur Identifizierung des Typs kritischer Lasten

Übersicht Beim Beulen einer Konstruktion unterscheidet man bekanntermaßen zwei verschiedene physikalische Phänomene. Das eine ist das Durchschlagproblem und das andere das Verzweigungsproblem. In der vorliegenden Arbeit wird ein Kriterium zur Identifikation dieser beiden Instabilitätstypen entwickelt. Es wird gezeigt, daß bei der allgemeinen Finite-Element-Analyse die Art der kritischen Punkte direkt durch die inkrementellen Gleichgewichtsbedingungen bestimmt werden kann: Sie hängt lediglich davon ab, ob das Lastinkrement bei der kleinsten kritischen Last Arbeit leistet.

     

A note on the critical points of equilibrium paths in imperfect structures

In a recent work (Int. J. Solids Struct. 37 (2000) 1561) by one of the authors, an extended system for calculating critical points of equilibrium paths in imperfect structures was presented. However, the extremum nature of these points was not analyzed explicitly in that paper. In this note, we will fill in the gap and establish a sufficient condition for determining the buckling strength of imperfect structures.     

The effect of entanglements on the rheological behavior of polybutadiene critical gels

We investigated the stress relaxation behavior of critical gels originating from six nearly monodisperse, highly entangled polybutadiene melts of different molecular weight from 18000 to 97 000 g/mole. The polymers were vulcanized by a hydrosilation reaction which takes place nearly exclusively at the pendant 1,2-vinyl sites distributed randomly along the polybutadiene chain. The BSW spectrum represents the relaxation of the initial uncrosslinked precursor. A characteristic parameter is the longest relaxation time of the precursor. Crosslinking increases this longest time even further. Surprisingly, the relaxation spectrum of the precursor is not altered much by the crosslinking except for an additional long time behavior. At the gel point (critical gel), this long time behavior is self-similar. It follows the typical power law as described by the Chambon-Winter gel equation, G(t) = St ^–n , in the terminal zone. The critical relaxation exponent was found to be close to n = 0.5 over a range of stoichiometric ratios and for all precursor molecular weights analyzed. A new scaling relationship was found between the gel stiffness, S, and the precursor molecular weight of the form: S M _w ^zn , where exponent z from the zero shear viscosity-molecular weight relationship, ₀ M _w ^z , is commonly found to be z = 3.3 – 3.6.     

Imperfection sensitivity of hilltop branching points of systems with dihedral group symmetry

Imperfection sensitivity of a hilltop branching point occurring as a coincidence of a limit point and a double bifurcation point of a finite-dimensional, elastic, conservative system equivariant to the dihedral group is investigated. In the neighborhood of this point, the potential is expanded into a power series of independent state variables, loading parameter and imperfection magnitude. The form of the expansion is determined through exploitation of dihedral-group symmetry. For the perfect system, the hilltop branching point and bifurcated paths are shown to be all unstable. For an imperfect system, equilibrium paths in general break into a series of paths: including fundamental, complementary and aloof paths. The imperfection sensitivity laws for maximum (critical) points of loading on these paths are obtained as a novel finding of this paper. Critical points on the fundamental and complementary paths enjoy a piecewise linear law, which is less severe than a one-half or two-thirds power law for the double bifurcation point. By contrast, maximum points on aloof paths suffer more severe sensitivity. The hilltop branching point thus displays complex system of imperfection sensitivities. As numerical examples, imperfection sensitivity of simple structural models with the hilltop point is investigated to ensure the validity of the present formulation.     

Bifurcation problem of critical points for quadratic differential systems

1.IntroductionWeconsiderthequadraticdifferentialsystemoftype(II)and(ill):.z~~y sx IXZ mxg g','~x(l ox)(1.1)f~~g 6x lx' mxy ny','~x(l ax by)(1.2)Asacontributionofsolvingbifurcationproblemofquadraticsystems,westudyinthispaperbifurcationofcriticalpoints(CPs)…     

On the dynamical symmetry points and the orientations of the principal axes of inertia of a rigid body

For an arbitrary rigid body, all dynamical symmetry points are found, and the directions of the axes of dynamical symmetry are determined for these points. We obtain conditions on the principal central moments of inertia under which the Lagrange and Kovalevskaya cases can be realized for the rigid body. We also analyze the set of orientations of the bases formed by the principal axes of inertia for various points of the rigid body.     

The effect of swirl on critical heat flux in annular two-phase flow

The adaptation of a calculation method for critical heat flux using an annular flow model is described. If the swirl is assumed to deposit all the entrained drops onto the liquid film and to persist for a certain distance after the swirl device, good comparisons with experiment are obtained. A simple method for calculating the length over which the swirl persists is given.     

一种提高极值点处收敛精度的三阶WENO-Z格式

为了提高三阶WENO-Z格式在极值点处的计算精度,通过理论推导给出三阶WENO格式满足收敛精度的充分条件。采用泰勒级数展开的方式,推导给出所构造格式非线性权重的计算公式,并综合权衡计算精度和计算稳定性确定所构造格式的参数。通过两个典型的精度测试,验证了改进格式在光滑流场极值点区域逼近三阶精度。进一步选用激波与熵波相互作用和Richtmyer-Meshkov不稳定性等经典算例,证实了本文提出的改进格式WENO-PZ3相较其他格式（WENO-JS3和WENO-Z3）不仅具有较高的精度,而且降低了格式的耗散,提高了对流场结构的分辨率。     

The effect of biaxial loading on the critical resolved shear stress of zinc single crystals

This paper reports an experimental investigation to test the hypothesis that the critical resolved shear stress is independent of the resolved normal stress on the active slip system plane. The resolved normal stress is varied independently for crystals of constant orientation by applying a uniform biaxial tension stress through slitted rubber strips glued to a flat tension sample. It is found that the critical resolved shear stress is constant until the resolved normal stress on the active slip system becomes twice the resolved shear stress; and then, for further increases in the resolved normal stress, the critical resolved shear stress begins to decrease.     

A nonlinear symmetry breaking effect in shear cracks

Shear cracks propagation is a basic dynamical process that mediates interfacial failure. We develop a general weakly nonlinear elastic theory of shear cracks and show that these experience tensile-mode crack tip deformation, including possibly opening displacements, in agreement with Stephenson's prediction. We quantify this nonlinear symmetry breaking effect, under two-dimensional deformation conditions, by an explicit inequality in terms of the first and second order elastic constants in the quasi-static regime and by semi-analytic calculations in the fully dynamic regime. Our general results are applied to various materials. Finally, we discuss related works in the literature and note the potential relevance of elastic nonlinearities for various problem, including frictional sliding.     

Regularization and symmetry breaking for the thermistor problem with the Thomson effect

We examine a mathematical model of the thermistor taking into account the Thomson effect. We show that this system of equations is always uniquely solvable, in sharp contrast with the case where the Thomson effect is neglected. Another feature of the problem is breaking of the symmetry of the solutions in accordance with practical experiments.

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Sommario Si esamina un modello matematico per il dispositivo noto come termistore tenendo conto anche dell'effetto Thomson. Viene provato che il problema al contorno per il sistema di equazioni differenziali ordinario che regge il dispositivo ha sempre una e una sola soluzione a differenza di quanto accade quando l'effetto Thomson è transcurato. Si prova inoltre la perdita di simmetria della soluzione, fatto rilevato anche sperimentalmente.

     

A study of flow properties near critical points in the near wake of a circular cylinder

By applying the phase-plane technique to velocity data in the near-wake of a circular cylinder, three types of critical points are identified. Foci and saddle points occur most frequently, but a significant number of nodes is also found. Flow topology and properties associated with these points are examined in some detail. While foci and saddle points are associated with maxima of local vorticity and strain rate respectively, nodes are associated with a strong local divergence, indicating significant local three-dimensionality. The relative probability of time delay between critical points is also discussed.     

A constitutive model for the Mullins effect with changes in material symmetry

When an unfilled or particle reinforced rubber is subjected to cyclic loading–unloading with a fixed amplitude from its natural reference configuration, the stress required on reloading is less than on the initial loading for a deformation up to the maximum value of the stretches achieved. The stress differences in successive loading cycles are largest during the first and second cycles and become negligible after about 4–6 cycles. This phenomenon is known as the Mullins effect. In this paper new experimental data are reported showing the change in material symmetry for an initially undamaged and isotropic material subjected to uniaxial and biaxial extension tests. The effect of preconditioning in one direction on the mechanical response when loaded in a perpendicular direction is discussed. A simple phenomenological model is derived to account for stress softening and changes in material symmetry. The formulation is based on the theory of pseudo-elasticity, the basis of which is the inclusion of scalar variables in the energy function. When active, these variables modify the form of the energy function during the deformation process and therefore change the material response. The general formulation is specialized to pure homogeneous deformation in order to fit the new data. The numerical results are in very good agreement with the experimental data.     

On the Noether symmetry and Lie symmetry of mechanical systems

The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates. The Lie symmetry is an invariance of the differential equations of motion under the transformations. In this paper, the relation between these two symmetries is proved definitely and firstly for mechanical systems. The results indicate that all the Noether symmetries are Lie symmetries for Lagrangian systems meanwhile a Noether symmetry is a Lie symmetry for the general holonomic or nonholonomic systems provided that some conditions hold. The project supported by the National Natural Science Foundation of China (19972010)     

Slip boundary effect on the critical Reynolds number of subcritical transition in channel flow

In this letter, the effect of slip boundary on the origin of subcritical transition in two-dimensional channel flows is studied numerically and theoretically. It is shown that both the positive and the negative slip lengths will increase the critical Reynolds number of localized wave packet and hence postpone the transition. By applying a variable transformation and expanding the variables about a small slip length, it is illustrated that the slip boundary effect only exists in the second and higher order modulations of the no-slip solution, and hence explains the power law found in simulations, i.e. the relative increment of the critical Reynolds number due to the slip boundary is proportional to the square of the slip length.     

The symmetry group of gradient sensitive fluids

Gradient sensitive fluids, for which there is constitutive function which depends both on the density and its spatial gradient, are seen as a special subclass of elastic materials of grade two and characterized through an appropriate symmetry group, which is explicitly described and discussed.