Three-dimensional studies on bicomponent extrusion

The present work is concerned with the mathematical modelling and numerical simulation of three-dimensional (3-D) bicomponent extrusion. The objective is to provide an understanding of the flow phenomena involved and to investigate their impact on the free surface shape and interface configuration of the extruded article. A finite element algorithm for the 3-D numerical simulation of bicomponent stratified free surface flows is described. The presence of multiple free surfaces (layer interface and external free surfaces) requires special free surface update schemes. The pressure and viscous stress discontinuity due to viscosity mismatch at the interface between the two stratified components is handled with both a double node (u–v–w–P ₁ –P ₂ –h ₁ –h ₂) formulation and a penalty function (u–v–w–P–h ₁ –h ₂) formulation.The experimentally observed tendency of the less viscous layer to encapsulate the more viscous layer in stratified bicomponent flows of side-by-side configuration is established with the aid of a fully 3-D analysis in agreement with experimental evidence. The direction and degree of encapsulation depend directly on the viscosity ratio of the two melts. For shear thinning melts exhibiting a viscosity crossover point, it is demonstrated that interface curvature reversal may occur if the shearing level is such that the crossover point is exceeded. Extrudate bending and distortion of the bicomponent system because of the viscosity mismatch is shown. For flows in a sheath-core configuration it is shown that the viscosity ratio may have a severe effect on the swelling ratio of the bicomponent system.Modelling of the die section showed that the boundary condition imposed at the fluid/fluid/wall contact point is critical to the accuracy of the overall solution.     

Estimate of the Latent Free Energy and Damage at the Tip of an Opening–Mode Crack

A method of estimating the latent elastic energy associated with the microinhomogeneity of the stress and plastic–strain fields inside the plastic zone localized near the tip of an opening–mode crack (Dugdale zone) under conditions of plane stresses is proposed. The microinhomogeneity of plastic flow upon small strain hardening is taken into account only in the form of considerable distortion of the geometry of the free surfaces of the plastic zone. The damage that developes because of release of the latent free energy is estimated depending on the magnitude of the crack opening.     

Oscillations of an ideal liquid acted upon by subface-tension forces. Case of a doubly connected free surface

Many articles have appeared on the problems of small oscillations of an ideal liquid acted upon by surface-tension forces. Oscillations of a liquid with a single free surface are treated in [1, 2]. Oscillations of an arbitrary number of immiscible liquids bounded by equilibrium surfaces on which only zero volume oscillations are assumed possible are investigated in [3], We consider below the problem of the oscillations of an ideal liquid with two free surfaces on each of which nonzero volume disturbances are kinematically possible. The disturbances satisfy the condition of constant total volume. A method of solution is presented. The problem of axisymmetric oscillations of a liquid sphere in contact with the periphery of a circular opening is considered neglecting gravity. The first two eigenfrequencies and oscillatory modes are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 64–71, May–June, 1976.In conclusion, the author thanks F. L. Chernous'ko for posing the problem and for his attention to the work.     

A plane steady-state free-boundary problem for the navier-stokes equations

The model problem of the plane slow steady-state motion of a viscous incompressible fluid with a free boundary is investigated. It is assumed that the free boundary does not have any points in common with the solid surfaces confining the fluid. By the solution of the auxiliary fixed-boundary problem for the Navier-Stokes equations the problem is reduced to an operator equation describing the form of the free surface. The existence and uniqueness problems for the solution and its qualitative behavior are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 91–102, May–June, 1972.In conclusion, the authors would like to thank R. M. Garipov and V. Kh. Izakson for affording an opportunity to become acquainted with the results of their unpublished work.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

Natural convection in a vertical cylindrical vessel with heat supplied to its side and free surfaces

A study was made in [4] of convective flow and heat transfer in a vertical cylindrical vessel with heat supplied to the free surface of the fluid, using the numerical simulation method. The results of this study were obtained for a comparatively short heating time for a fixed ratio of heat fluxes to the side and free surfaces. The present study is a continuation of the one in [4]. By using a more precise numerical simulation method, calculations could be made over a fairly wide range of determining parameters, and results were obtained for both short and long heating times.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 51–56, November–December, 1984.     

Penetrative convection in the isothermally incompressible fluid approximation

The onset of penetrative convection in an infinite horizontal fluid layer bounded by isothermal rigid or free nondeformable surfaces is numerically examined. It is assumed that the specific volume of the fluid depends quadratically on temperature and reaches a minimum inside the layer. The isothermally incompressible fluid convection model in which, as distinct from the Oberbeck-Boussinesq approximation, the thermal expansion is not assumed to be small is considered. Both the neutral stability curves of the conductive regime and the amplitudes of two-dimensional periodic and three-dimensional doubly-periodic convective flow are calculated. The results are compared with those previously obtained for the equations of penetrative convection in the Boussinesq approximation.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 40–52, March–April, 1996.     

Nonlinear equations of elastic deformation of plates

A method for constructing nonlinear equations of elastic deformation of plates with boundary conditions for stresses and displacements at the face surfaces in an arbitrary coordinate system is proposed. The initial three–dimensional problem of the nonlinear theory of elasticity is reduced to a one–parameter sequence of two–dimensional problems by approximating the unknown functions by truncated series in Legendre polynomials. The same unknowns are approximated by different truncated series. In each approximation, a linearized system of equations whose differential order does not depend on the boundary conditions at the face surfaces which can be formulated in terms of stresses or displacements is obtained.     

Effect of blowing on supersonic dust-laden gas flow round a blunt body

One of the main problems which arise in the design of high-speed aircraft is the protection of the streamline surfaces against the erosion effect of solid particles and drops occurring in the free stream. For this purpose it is possible to use the device of blowing cold gas. This leads to the formation of a boundary layer of high density in which the particles are decelerated [1]. The present study investigates the effectiveness of this method of erosion protection in the example of supersonic flow round a sphere.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 178–181, March–April, 1986.     

Method of generalized similitude in free convection problems with arbitrary distribution of the temperature or heat flux on a vertical wall

The self-similar problem of free convection near a heated vertical plate was solved for the first time in [1] for the simplest case of a constant wall temperature. In [2], Yang proved the existence of a self-similar solution to the problem of free convection for vertical plates and cylinders on the surfaces of which the temperature has a power-law distribution. In [3], Yang's solution was generalized to the case of free convection near a slender figure of revolution, but also only in the self-similar case of a power-law distribution of the temperature on the wall. In [4], this problem was solved in an extended nonsimilar formulation but by an artificial and not general method similar to Gertler's, the convergence of the approximations being slow. The present paper contains the solution to the problem of free convection near a vertical plate with arbitrary distribution of the temperature or heat flux on its surface. Rigorous application of the method of generalized similitude [5] leads in this case to universal equations that present insuperable computational difficulties, which forces one to use a simplified but fairly general method to solve this class of problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 167–170, May–June, 1980.I thank L. G. Loitsyanskii and E. M. Smirnov for discussing the results and for valuable comments.     

Stability of flows with discontinuity surfaces

A study is made in the linear formulation of flows with homogeneous distribution of the parameters in expanding regions separated by boundaries that are either discontinuity surfaces of an arbitrary nature or surfaces with effective boundary conditions. Examples of such flows are the decay of an arbitrary discontinuity [1] and flow in a tube with a region of heat release [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 10–18, September–October, 1982.We thank A. G. Kulikovskii for helpful discussions.     

Hydrodynamics of wetting

Fluid motion along a smooth, solid surface is examined when the free surface forms a final visible angle with the solid boundary. The dependence of this angle on the velocity with allowance for capillary forces is determined. The Reynolds number is small. The problem of the motion of the phase interface in a capillary and the spreading of viscous fluid drops on solid surfaces are solved. Experimental results are explained. Up to now, not only were analytical results lacking in this field, but also there was not even a precise formulation of the problem (see the review in [1]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 76–84, September–October, 1976.     

Nonstationary waves in the continuously stratified flow of a fluid of finite depth

A theoretical investigation is made of the development of linear two-dimensional waves in a continuously stratified flow of an ideal incompressible fluid. The waves are generated by pressures that are independent of time and that are applied at time t=0 to a bounded region on the free surface of an initially undisturbed flow. The stationary internal waves generated by such a disturbance have been investigated in [1–3]. The nonstationary waves in a continuously stratified fluid that are generated by initial disturbances or periodic surface pressures applied to the entire free surface have been studied in [4–7] in the absence of a slow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 87–93, November–December, 1976.     

Effects of partial crack–face contact for the bending of thin shell structures

The physical occurrence that crack surfaces are in contact at the compressive edges when a flat or a shell is subjected to a bending load has been recognized. This article presents a theoretical analysis of crack–face contact effect on the stress intensity factor of various shell structures such as spherical shell, cylindrical shell containing an axial crack, cylindrical shell containing a circumferential crack and shell with two non-zero curvatures, under a bending load. The formulation of the problem is based on the shear deformation theory, incorporating crack–face contact by introducing distributed force at the compressive edge. Material orthotropy is concerned in this analysis. Three-dimensional finite element analysis (FEA) is conduced to compare with the theoretical solution. It is found that due to curvature effect crack–face contact behavior in shells differs from that in flat plates, in that partial contact of crack surfaces may occur in shells, depending on the shell curvature and the nature of the bending load. Crack–face contact has significant influence on the stress intensity factor and it increases the membrane component but decreases the bending component.     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.     

Combining photoelasticity and finite-element methods for stress analysis using least squares

Photoelastic data are combined with the finite-element method for stress solutions over regions partially bounded by free surfaces and axes of symmetry. Least-squares solutions are obtained without presumed values of applied forces at element nodes and without isoclinic data. Varied example problems are used to compare the results to independent photoelastic and finite-element solutions and to theoretical stress values.Paper was presented at the 1989 SEM Spring Conference on Experimental Mechanics held in Cambridge, MA on May 28–June 1.     

Forces and moments acting on bodies rotating about a symmetry axis in a free molecular flow

Analytic expressions are obtained for the forces and moments acting on symmetrically rotating convex figures of revolution moving in a free molecular flow of rarefied gas under the following assumptions: the velocity distribution function of the molecules of the oncoming flow is Maxwellian and the incident molecules have a diffuse—specular interaction with the surface of the body. For bodies with arbitrary piecewise smooth generator, general expressions are found in terms of quadrature for the components of the aerodynamic forces and moments. For a disk, sphere, and cylindrical and conical surfaces, the integration of the forces and the moments, which depend on the rotation of the body, is carried out to the end. For the moments of the forces, graphs are plotted of the errors of the hypothermal approximation as a function of the velocity ratio.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 151–155, May–June, 1980.     

Transport phenomena in a knudsen molecular gas in a magnetic field

We consider the behavior of a strongly rarefied (Knudsen) polyatomic gas between two surfaces that have different temperatures in a magnetic field. We show that in the magnetic field H there can arise a flux of gas and a heat flux along the surfaces (odd functions in H), and also normal and tangential forces on the walls.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 124–130, January–February, 1979.     

Shear flows of a liquid with a power-type rheological law in the presence of a constant transverse velocity component

Plane shear flows of dilatant liquids are considered in the presence of a constant transverse velocity component of the liquid; this component arises from uniform injection (or suction) at the bounding surfaces. It is shown that the presence of transverse drift in the liquid influences the rate of propagation of the leading edge of the shear waves, and this has a number of effects on the flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 15–21, July–August, 1973.     

The zonal linearization method for multidimensional mass transfer problems in porous media

A number of methods have been proposed in recent years for calculating the combined flows of immiscible and miscible liquids in strata to systems of boreholes. We propose a method which can naturally be called the zonal linearization method [1]. It is more compact than the usual finite-difference method and has high accuracy, in particular, in the neighborhood of a borehole, since it is closely similar to the method of characteristics. The method can be applied to both continuous and discontinuous flows and in principle makes it possible to investigate the formation and breakdown of discontinuities. As distinct from the method of characteristics, it is well suited to programming and implementation on a computer, and it also makes it possible to obtain an approximate analytic solution of the problem in many cases and to estimate the accuracy of the solution. The method is based on the zonal linearization of the equation for mass conservation in the total flow between chosen surfaces or contour lines (lines of equal saturation or concentration). Determination of the dynamics of the contour surfaces leads to a Cauchy problem for a system of integrodifferential equations involving partial derivatives. The zonal linearization method is a development of the scheme described in [2–4], and the method of solving the Cauchy problem is a generalization of the methods described in [4–13]. The essence of the method and its convergence are illustrated by means of two-dimensional problems in two-phase filtration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–80, July–August, 1973.