On the role of symmetry in Saint-Venant’s problem

In the relaxed Saint-Venant’s elastic problem, in virtue of Saint-Venant’s Postulate, the pointwise assignments of tractions at cylinder plane ends are replaced by the assignments of the corresponding resultant forces and moments. The solution indeterminacy so introduced is traditionally overcome by postulating that some specific features characterize the elastic state. In this work a relaxed incremental equilibrium problem is posed for a heterogenous anisotropic cylinder, whose tangent elasticity tensor field possesses the usual major and minor symmetries, is positive definite, independent from the axial coordinate and endowed with a plane of elastic symmetry orthogonal to the cylinder axis. Symmetry has been consistently employed to formulate the basic problems of extension, bending, torsion and flexure as symmetric and antisymmetric problems respectively. It is shown that Saint-Venant’s postulate, momentum balance and symmetry are sufficient, without resorting to any a priori assumption, to determine the general form of the displacement field and to remove the solution indeterminacy.     

Bending equation for a quasianisotropic plate

In the framework of the linear theory of elasticity, an exact bending equation is obtained for the median plane of a plate whose material is a monoclinic system with the axis of symmetry perpendicular to the plate plane. As an example, the equation of the median plane of an isotropic plate is considered; the operator of this equation coincides with the operator of Sophie Germain’s approximate equation. As the plate thickness tends to zero, the right-hand side of the equation is asymptotically equivalent to the right-hand side of the approximate equation. In addition, equations relating the median plane transverse stresses and the total stresses in the plate boundary planes to the median plane deflexions are obtained.     

Stereoscopic multi-planar PIV measurements of in-cylinder tumbling flow

The non-reacting flow field within the combustion chamber of a motored direct-injection spark-ignition engine with tumble intake port is measured. The three-dimensionality of the flow necessitates the measurement of all three velocity components via stereoscopic particle-image velocimetry in multiple planes. Phase-locked stereoscopic PIV is applied at 15 crank angles during the intake and compression strokes, showing the temporal evolution of the flow field. The flow fields are obtained within a set of 14 axial planes, covering nearly the complete cylinder volume. The stereoscopic PIV setup applied to engine in-cylinder flow and the arising problems and solutions are discussed in detail. The three-dimensional flow field is reconstructed and analyzed using vortex criteria. The tumble vortex is the dominant flow structure, and this vortex varies significantly regarding shape, strength, and position throughout the two strokes. The tumble vortex center moves clockwise through the combustion chamber. At first, the tumble has a c-shape which turns into an almost straight tube at the end of the compression. Small-scale structures are analyzed by the distribution of the turbulent kinetic energy. It is evident that the symmetry plane only represents the 3D flow field after 100 CAD. For earlier crank angles, both kinetic energy (KE) and turbulent kinetic energy (TKE) in the combustion chamber are well below the KE and TKE in the symmetry plane. This should be taken into account when the injection and breakup of the three-dimensional fuel jet are studied. The mean kinetic energy is conserved until late compression by the tumble motion. This conservation ensures through the excited air motion an enhancement of the initial air-fuel mixture which is of interest for direct-injection gasoline engines.     

On the radiation pattern of a paraboloid of revolution

Summary The radiation pattern of a paraboloid of revolution is determined by using the mathematical form of Huygens' principle where the field intensities in a given domain are expressed in their tangential components on the boundary. These tangential components are supposed to have the values which follow from the reflection of a plane wave at an infinite metal plate, coinciding with the tangent plane to the surface of the paraboloid. Expressions involving infinite series of Bessel functions are obtained which give the intensities of the distant field when an electric dipole is placed at the focus. Approximations are made for the main lobe in the planes perpendicular to and through the dipole. Finally Zuhrt's method leads to the same results as the approximations given in the present paper.     

Space and space-time longitudinal velocity correlations in the turbulent far wake of a flat plate in incompressible flow

This paper presents longitudinal space and space-time conventional velocity correlations measured in the wake of a two-dimensional flat-plate at zero incidence, with fully-turbulent boundary layers at its sharp trailing edge. Iso-correlation contour plots are given in the plane of symmetry and normal to this plane in the assumed asymptotic (in the mean velocity sense) part. Comparisons with initially laminar 2-D wake results show that the classical Double-Roller-Eddy model developed for initially laminar wakes is not adapted to initially turbulent wakes; on the other hand, comparison with supersonic wake of the same nature seems to confirm this conclusion. The lack of detailed experimental data in this field is outlined, together with the need of a refined theoretical approach to the coherent structures existing in such flows.     

Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.     

应变损伤材料平面应力动态裂纹尖端场的渐近解

采用塑性动力学方程,对应变损伤材料的平面应力动态裂纹尖端场进行了渐近分析。假定损伤规律服从反比例关系,对平面应力问题,导出了本构方程,并给出了动态弹塑性场的渐近解,揭示了场的渐近特性。     

Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory

Several results are presented concerning symmetry properties of the tensor of third order elastic moduli. It is proven that a set of conditions upon the components of the modulus tensor are both necessary and sufficient for a given direction to be normal to a plane of material symmetry. This leads to a systematic procedure by which the underlying symmetry of a material can be calculated from the 56 third order moduli. One implication of the symmetry conditions is that the nonlinearity parameter governing the evolution of acceleration waves and nonlinear wave phenomena is identically zero for all transverse waves associated with a plane of material symmetry.     

Acoustoelasticity: Ultrasonic stress field reconstruction

Based on the theory of acoustoelasticity, a new ultrasonic stress reconstruction method—the generalized acoustic ratio (GAR) technique—is developed for locally plane structures and orthotropic materials. For given transit times of the three wave modes and the shear wave polarization angle, the local plane stress tensor is uniquely determined. The GAR technique yields accurate stress estimates with relatively small temperature sensitivity. Based on calibration constants from three uniaxial specimens, the entire stress field in a compact tension specimen is reconstructed. The results are in very good agreement with stress predictions from an elastoplastic finite element analysis. To further improve the measurements, a numerical technique, the stress field approximation (SFA) technique, is developed. The SFA technique uses a smooth local bicubic spline approximation and aims at improving the overall stress field estimate by enforcing the equilibrium equations, the stress boundary conditions and symmetry conditions. Numerical results show that both the average error and its spread are indeed reduced.     

Flow structures in initial region of two interacting parallel plane jets

The initial region of two interacting parallel plane jets with a truncated bluff body at the plane of symmetry is studied with velocity measurements in the frequency and time domains. The mean flows of the two plane jets are nearly parallel to the plane of symmetry, and their inner and outer mixing regions are found. Within these mixing regions their respective trains of coherent structures, which are of vortical form, are established. The inner vortices are inwardly rotational, while the outer vortices rotate outwardly. The successive initial vortices in the inner mixing region seem to undergo two processes of either pairing or amalgamation. The former results in the formation of a nearly circular coherent structure, while the latter results in the elongated structure. Both the inner and other structures experience fairly rapid decay. The process of decay also seems to be either the usual decay or the division of the coalesced structure back into the individual vortical structures followed by their decay.     

Equilibrium shapes of a drop pendant in an electrostatic field

The stationary shapes of a conducting fluid drop in the gap between the plates of a plane capacitor are studied. The drop is held on the upper plate by the surface tension forces. The self-consistent problem of the determination of the drop shape and the charge distribution over its surface is solved. Estimates are obtained for the maximum volume of the stationary drop at the given fluid parameters and electric field strength.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation

A non-linear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section is developed. Non-linear in-plane and out-of-plane warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, together with the assumption that the cross-section is undeformable in its own plane, the non-linear warping is described in terms of the flexural and torsional curvatures. Due to the internal constraints, the displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. Three non-linear differential equations of motion up to the third order are derived using the Hamilton principle. Taking into account the order of magnitude of the various terms, the equations are simplified and the importance of each contribution is discussed. The effect of symmetry properties is also outlined. Finally, a discrete form of the equations is given, which is used in Part II to study dynamic coupling phenomena in conditions of internal resonance.     

Form invariance and Lie symmetry of variable mass nonholonomic mechanical system

IntroductionThereisacloserelationbetweenthesymmetryandtheconservedquantityinamechanicalsystem .ModernmethodstofindconservedquantityofamechanicalsystemaremainlyNoethersymmetrymethod[1]andLiesymmetrymethod[2 ].NoethersymmetryisaninvarianceoftheHamiltonactionundertheinfinitesimaltransformations.Liesymmetryisaninvarianceofthedifferentialequationsundertheinfinitesimaltransformations.Inthepasttenyears,aseriesofimportresultshavebeenobtainedonthestudyoftheNoethersymmetryandLiesymmetry[3～12 ].Thefo…     

Charged jet of an incompressible liquid in an electric field

The problem of the jet flow of an incompressible liquid with free boundaries in an electric field is solved in the approximation of a laminar boundary layer. An exact solution for a round jet is found in the class of self-similar solutions. In the case of a flat slit jet, a solution is constructed in the form of a series in powers of the coordinate transverse to the plane of symmetry. The dependence of the radius (half-width) on the longitudinal coordinate is given. Branch of the Karpov Physicochemistry Institute, State Science Center, Obninsk 249020. Karpov Physicochemistry Institute, State Science Center, Moscow 115523. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 12–16, July–August, 1998.     

压电材料平面应力状态的直线裂纹问题一般解

研究了含直线裂纹系的压电材料平面应力问题单个裂纹和双裂纹问题的封闭解答表明，在裂纹尖端，应力、电场强度和电位移有１／２阶的奇异性并与前人结果比较了产生电场奇异性的物理因素     

压电材料平面应力状态的直线裂纹问题一般解

研究了含直线裂纹系的压电材料平面应力问题单个裂纹和双裂纹问题的封闭解答表明，在裂纹尖端，应力、电场强度和电位移有１／２阶的奇异性并与前人结果比较了产生电场奇异性的物理因素     

FORM INVARIANCE AND LIE SYMMETRY OF THE GENERALIZED HAMILTONIAN SYSTEM

The form invariance and the Lie symmetry of the generalized Hamiltonian system are studied. Firstly, de?nitions and criteria of the form invariance and the Lie symmetry of the system are given. Next, the relation between the form invariance an…     

The inclusion problem of an infinite strip

Summary This paper presents a direct solution for the stresses in an infinite strip containing a rigid circular inclusion located at any position within the strip. The solution is applicable to any stress system even with respect to the cross axis of symmetry of the strip. The strip is regarded in a state of generalized plane stress.Numerical examples are shown corresponding to a certain given size and location of the inclusion. Both the cases of longitudinal tension and transverse bending are considered. The parametric coefficients involved in each case are determined. The hoop-stress in the strip at the surface of inclusion as well as the normal stress along the cross axis of symmetry of the strip are evaluated.     

Tomographic reconstruction of internal wave patterns in a paraboloid

Using tomographic synthetic schlieren, we are able to reconstruct the three-dimensional density field of internal waves. In this study, the waves are radiating from an oscillating sphere positioned eccentrically at the surface of a paraboloidal domain filled with a uniformly stratified fluid. We find that the prediction by ray tracing corresponds well with the observed intensities of the wave field. Remarkably, for a specific value of the forcing frequency, we observe convergence of internal wave energy to an internal wave attractor. The attractor is found to dominate fluid motion in the plane perpendicular to the plane spanned by the symmetry axis and the oscillator position.     

Focusing of weak shock waves in confined axisymmetric chambers

Theoretical study of a weak shock wave focusing process on a spherical region in confined 3-D axisymmetric chambers is presented. The chambers are elliptic or parabolic in the plane cross-section containing their axis of symmetry. In the elliptic case a spherical shock wave of constant strength generated at one of the focal points will reflect off the chamber wall and converge on a spherical region around the second focus of the chamber. It is shown that the pressure distribution on the converging spherical shock wave is not homogeneous. In the parabolic case two possibilities of shock generation are considered. In the first one a plane shock wave of constant intensity is send in the inner of the chamber. This shock wave with the plane perpendicular to the symmetry axis will after the reflection off the chamber wall transform to a spherical shock with non-homogeneous pressure distribution. Alternatively, a spherical shock of constant intensity generated at the focus of the paraboloidal chamber will after the reflection transform to a plane shock with non-homogeneous pressure distribution propagating in the outer of the chamber. The above mentioned problems are solved within the frame of the geometrical acoustics approximation and the flow fields as well as the non-uniform shock strengths behind the converging wave fronts are calculated.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.