Three-dimensional isothermal lava flows over a non-axisymmetric conical surface

Within the thin-layer approximation for a highly-viscous heavy incompressible fluid, a hydrodynamicmodel of a 3D isothermal lava flow over a non-axisymmetric conical surface is constructed. Using analytical methods, a self-similar solution for the law of leading-edge propagation is obtained in the case of a flow from a non-axisymmetric source located at the apex of a conical surface with smoothly varying properties. In the case of a flow over a substantially non-axisymmetric surface, it is shown that there exists a self-similar solution for the free-surface shape and the law of leading-edge motion. This solution is studied numerically for particular examples of the substrate surface and the source. In the general case of a non-self-similar flow over a substantially non-axisymmetric conical surface, a local analytical solution is obtained for the free-surface shape and the velocity field near the leading flow front.     

Draft of a nozzle with the circulating outflow of a vortical flow of gas

The presence of circulation in an outflowing gas leads to a change in the working parameters of a nozzle. The question of the mass flow rate and the draft of a nozzle without a diffusor (a point) for twisted flows has been studied theoretically and experimentally [1–6]. The use of nozzles with a supersonic part introduces a considerable degree of complication into the method for the analytical calculation of the draft characteristics and the program for their experimental investigation. In [2, 7], a theory of a nozzle is formulated for a model of a potential circulating flow of gas; in [5, 8], an electronic computer was used to solve the complete system of the equations of gasdynamics for the motion of a rotating flow along a nozzle; in [7, 9], an investigation was made of a variational problem of the shaping of a diffusor for a circulation flow. The calculation of the draft, carried out in the above-mentioned communications (with the exception of [2], in which a study was made of a partial model of an eddyless rotational motion), is bound up with labor-consuming computer calculations. In the present article, in a development of [3, 6], a quasi-one-dimensional theory of a supersonic nozzle for a vortical flow of gas is formulated and verified experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–149, September–October, 1975.     

Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.     

Configurations of gas-liquid two-phase bubbles in immiscible liquid media

The configuration of a “two-phase bubble” constituted of a gas phase and a liquid phase in an immiscible liquid medium is classified into three types: complete engulfing of a gas bubble inside a liquid shell, partial coalescence of a gas bubble and a liquid drop forming a three-phase contact line, and non-coalescence whereby a gas bubble and a liquid drop remain separated. Simple criteria have been presented by which the favorable type of configuration in a given system is predicted from the values of the spreading coefficients characterizing the system. Experiments using some combinations of liquids as well as air suggest the general validity of the criteria.     

Motion of a solidcontaininga spherical damper

For the purpose of modeling the motion of a solid with a cavity filled with a viscous fluid, M. A. Lavrent'ev [1] has proposed a model in the form of a solid with a spherical cavity in which another solid, spherical in shape, is enclosed. The sphere is separated from the cavity walls by a small, clearance in which viscous forces act (a lubricating film). This simple model with a finite number of degrees of freedom possesses certain mechanical properties of a solid with a cavity containing a viscous fluid. Study of this model is therefore of interest.The present paper examines certain properties of the model, which will be termed a solid with a damper. It is shown that for a highviscosity lubricant the motion of a solid with a damper can be described by the same equations which pertain to the motion of a solid with a spherical cavity filled with a high-viscosity fluid. Expressions relating the parameters of the systems are obtained. If these relations are fulfilled, the systems become mechanically equivalent.The steady motions of a free solid with a damper and their stability conditions are determined.These motions and stability conditions hold for a body with a cavity filled with a viscous fluid [2].     

MINIMAX THEOREM AND SADDLE POINT THEOREM WITHOUT LINEAR STRUCTURE

I.IntroductionFanKyinequalityisaveryusefultoolinnoolinearanalysiswhichasserts:LetXbeanonemptycompactconvexsetinatopologicallinearspaceandWbeareal-valuedfunctiondefinedonXxX.Assumethat(i)9(x,y)islowersemicontinuousinac;(n)9(x,y)isquasiconcaveiny.ThenthereisboaXsuchthatsHPg(xo'Y)$sZPp(x,x).SinceFanKyprovedtheinequalityin1972,variousg...,.li..tlon,havebeen;v..byseveralauthors(see[l-3]andreferencestherein).Thepreviousgeneralizationsallremainedtheconditions(i)-'9(x,y)beinglowersemicontinuous…     

Kinetic Stabilization¶of a Nonlinear Sonic Phase Boundary

We consider a system arising in the study of phase transitions in elastodynamics – a system of two conservation laws, in a single space dimension. The system has two hyperbolic regions with an elliptic zone in between. A phase boundary is a strong discontinuity in a solution, with left and right states belonging to different hyperbolic regions. We call such a solution a phase wave. We first address the Riemann problem for initial states close to a fixed sonic phase wave, in the genuinely nonlinear case. This problem is naturally underdetermined. We propose two essentially different types of Reimann problems: a sonic one, which is smooth, and a kinetic one, which is only Lipschitz-continuous. Both problems are well posed owing to a shared stability condition that is of a purely sonic nature. In the kinetic case we prove the global existence of solutions to the Cauchy problem for initial data having small variation and close to a sonic kinetic wave. The crucial issue is the interaction of the phase boundary with a small wave of the same mode. The introduction of a pertinent quantity, called here detonation potential, ensures a balance between ingoing and outgoing waves. The proof is based on a Glimm-type scheme; we define a potential, which includes the detonation potential, along the strong discontinuity, and this potential controls the outbreak of unusual shocks. Accepted: June 9, 1999     

Proposed slip line fields for indenting with rough plane and curved dies

Summary Theoretical solutions are presented for the plane strain deformation of a rigid-perfectly plastic solid for (a) indenting a plane surface with a partially rough plane rigid die; (b) crushing of a wedge by a rough plane rigid die; (c) indenting by a pair of opposed rough circular dies.     

Reduction of dimensions in random,elastic soil medium

The paper deals with an application of the plane strain analysis in a stochastic three-dimensional soil medium. In a framework of random elasticity theory, the geostatical state of stresses and the problem of a unit force acting in a statistically homogeneous half-space are considered. Only the modulus of elasticity is considered to be random and is modelled as a three-dimensional (3-D) homogeneous random field. As the result of imposed constrains due to the plane strain assumption the additional body and surface forces are induced. In order to determine them, additional equations must be introduced. The equations in a form of constrain relations are proposed in this paper. These equations are also valid for a case of uniformly distributed external loading.First, the two-dimensional (2-D) problem and its reduction to the uni-axial strain state, for the gravity forces and uniform, unlimited surface loading is considered. Then, it is generalised into a 2-D schematization of the 3-D state. Next, the problem of a unit force acting in a statistically homogeneous half-space is considered. For a 3-D state of stress and strain the resulting stresses are compared with those for a 2-D state. These stresses for the multidimensional state of strain and stress are presented as a sum of two components. The first one reflects plane strain state stresses and is given in a form of a 3-D random field. This term allows for incorporating a spatial, 3-D soil variability into a two-dimensional analysis. The second component can be treated as a correction term and it represents the longitudinal influence of a 3-D analysis.Some numerical results are presented in this paper. The proposed method can be regarded as a framework for further research aiming at application to a variety of geotechnical problems, for which the plane strain state is assumed.     

A semi-analytical approach to Biot instability in a growing layer: Strain gradient correction,weakly non-linear analysis and imperfection sensitivity

Many experimental works have recently investigated the dynamics of crease formation during the swelling of long soft slabs attached to a rigid substrate. Mechanically, the spatially constrained growth provokes a residual strain distribution inside the material, and therefore the problem is equivalent to the uniaxial compression of an elastic layer.The aim of this work is to propose a semi-analytical approach to study the non-linear buckling behaviour of a growing soft layer. We consider the presence of a microstructural length, which describes the effect of a simple strain gradient correction in the growing hyperelastic layer, considered as a neo-Hookean material. By introducing a non-linear stream function for enforcing exactly the incompressibility constraint, we develop a variational formulation for performing a stability analysis of the basic homogeneous solution. At the linear order, we derive the corresponding dispersion relation, proving that even a small strain gradient effect allows the system to select a critical dimensionless wavenumber while giving a small correction to the Biot instability threshold. A weakly non-linear analysis is then performed by applying a multiple-scale expansion to the neutrally stable mode. By applying the global conservation of the mechanical energy, we derive the Ginzburg–Landau equation for the critical single mode, identifying a pitchfork bifurcation. Since the bifurcation is found to be subcritical for a small ratio between the microstructural length and the layer׳s thickness, we finally perform a sensitivity analysis to study the effect of the initial presence of a sinusoidal imperfection on the free surface of the layer. In this case, the incremental solution for the stream function is written as a Fourier series, so that the surface imperfection can have a cubic resonance with the linear modes. The solutions indicate the presence of a turning point close to the critical threshold for the perfect system. We also find that the inclusion of higher modes has a steepening effect on the surface profile, indicating the incipient formation of an elastic singularity, possibly a crease.     

结构强度在线分析的光电复合法

本文将电测方法与光弹性方法结合在一起,导出了应用读数应变εd就可以获取准静态情况下结构的载荷P,危险截面位置和最大应力峰值σmax(X0Y0Z0)的理论计算方法,从而为结构的在线强度评价提供一种实验分析方法,文中导出的εd-t,P-t,k-t及σ－t诸曲线谱的相关性理论,不仅揭示了εd与未知载荷P,无量纲应力系数K及应力σ之间的关系,更重要的是这种分析思想与计算机技术结合在一起,可以达到设备强度自动报敬警的作用,作为一个实例,作者应用本理论成功地解决了某钢厂热轧线上一个粗轧机机架在线的应力强度评价难题。     

Scenarios of the onset of unsteady regimes in the problem of plane convective flow through a porous medium

The problem of plane convective flow through a porous medium in a rectangular vessel with a linear temperature profile steadily maintained on the boundary is considered. The onset of unsteady regimes is investigated numerically. It is shown that their onset scenarios depend on the vessel dimensions and the seepage Rayleigh number and may be as follows: the generation of stable and unstable periodic regimes as a result of a one-sided bifurcation, the generation of a stable periodic regime as a result of an Andronov-Hopf cosymmetric bifurcation, the formation of a chaotic attractor, the branching-out of a stable quasi-periodic regime from a point of a single-parameter family of steady-state regimes, and the generation of unstable periodic regimes as a result of disintegration of homoclinic trajectories. The specifics of most of the bifurcations mentioned above are attributable to the cosymmetry of the problem considered.     

Crack kinking in a piezoelectric solid

The intrinsic coupling between the mechanical and the electric fields assigns a uniquefeature for the fracture in a piezoelectric solid. We model the kink of a crack by continuousdistribution of edge dislocations and electric dipoles. The problem admits an approach based onthe Stroh formalism. A set of coupled singular integral equations are derived for the dislocationand electric dipole density functions associated with a kinked crack. Numerical results indicatethat the crack tends to propagate in a straight line under a tensile stress and a positive electricfield. For a crack subjected to the mixed mode mechanical loading, a superimposed positiveelectric field tends to reduce the kink angle. The influence of the non-singular T-stress-chargeparallel to a crack is also investigated. It is shown that a transverse tensile stress or a positivetransverse electric field will lead to further deviation of the kinked crack from the crackextension line.     

Some Results in the Dynamical Theory of Porous Elastic Bodies

This paper is concerned with the linear dynamic theory of elastic materials with voids. First, a spatial decay estimate of an energetic measure associated with a dynamical process is established. Then, a domain of dependence inequality associated with a boundary-initial-value problem is derived and a domain of influence theorem is established. It is shown that, for a finite time, a solution corresponding to data of bounded support vanishes outside a bounded domain.     

Numerical simulation of liquefied fuel spills: I. Instantaneous release into a confined area

The shallow-water equations in radial symmetry are solved numerically to simulate the collapse of a cylindrical liquid column into an area surrounded by a concentric dike. The following three subcases of this problem are considered: a liquid column collapsing onto a layer of the same liquid, a liquid column collapsing onto a solid surface, and a column of lighter liquid collapsing onto a heavier liquid (i.e. liquefied natural gas (LNG) spilled onto water). The results for the three categories are compared and the differences and similarities between them are analysed.     

An analysis of thickness effects in the Izod test

The Izod impact test is analyzed numerically using a polymer constitutive relation with material parameters qualitatively representative of a polycarbonate. The computations are full 3D transient analyses using explicit time integration and accounting for finite strains. The main purpose of the analyses is a comparison of the stress and strain fields that develop for the various specimen geometries that are used in practice, ranging from a specimen with a square cross-section to a specimen with a width about a quarter of that value. It is shown that the response varies from something close to a plane strain response to something close to a plane stress response. The results illustrate the effect of the stress–strain behavior of polymers, which involves attaining a stress peak, followed by softening and then by the gradual evolution of a very stiff response resulting from increasing network stiffness.     

The puzzle of whip cracking – uncovered by a correlation of whip-tip kinematics with shock wave emission

During whip cracking the whip-tip reaches a supersonic velocity for a period of about 1.2 ms, thereby emitting a head wave with a parabolic-shaped geometry. A detailed study of this mechanism which encompasses the motion analysis of the whip-tip as well as the determination of the local origin of the shock emission requires a sophisticated recording technique. A pre-trigger framing high-speed video camera system was used which was triggered by an acoustical sensor and synchronized with a pulsed copper-vapour laser. The phenomena were visualized by the direct shadowgraph method and recorded cinematographically as digital images at a frame rate of 9 kHz using a CCD-matrix with pixels. The resulting series of frames allowed, for the first time, (i) a reconstruction of the whip-tip trajectory, (ii) a determination of the tuft velocity and acceleration, (iii) a correlation of whip-tip kinematics with shock wave emission, and (iv) a motion analysis of the turning and unfolding mechanism of the tuft. The tuft at the whip-tip was accelerated within a distance of about 45 cm from a Mach number of to a maximum of , thereby reaching a maximum acceleration of 50,000 g. The shock is emitted at the moment when the cracker, arriving at the turning point of the lash, is rapidly turned around. After emission of the shock wav within a short distance of only 20 cm. Received 3 March 1997 / Accepted 21 July 1997     

Ideal gas flow behind a finite-amplitude shock wave

The equations of one-dimensional (with a plane of symmetry) adiabatic motion of an ideal gas are transformed to a form convenient for studying flows between a moving piston and a shock wave of variable intensity. The solution is found for the equations of a motion containing a shock wave which propagates through a quiescent gas with variable initial density and constant pressure. This solution contains four arbitrary constants and, in a particular case, gives an example of adiabatic shockless compression by a piston of a gas initially at rest.     

Une technique de suivi d'interfaces pour la modélisation de la cavitation par poche

A tracking method is presented for the modeling of partial and supercavitation. The velocity and pressure fields in the cavitating flow are computed by a Navier–Stokes solver using a pseudo-compressibility method. The cavity flow is computed from the velocity field by a tracking method based on a volume of fluid technique (VOF). The method is illustrated by several computations, two cases of partial cavitation on a hydrofoil and a case of a cavitating body emerging at a free surface.