钢铁工业中的自由边值问题

This paper considers two novel free boundary problems that emerge from modelling processes basic to steel manufacture.The first process concerns the spray cooling of hot steel sheet during the process of continuous casting. Here, an important practical consideration is the non-monotonicity of the measured heat transfer from the steel as a function of the steel temperature. In order to understand this phenomenon, a two-phase flow model is written down for the heating and vapourisation of the water spray. This model relies on a microscale analysis of droplet vapourisation and, in a steady state, it reduces to a coupled system of nonlinear ordinary differential equations for the spray temperature and water content. This system predicts the conditions for the existence or otherwise of a free boundary separating the two-phase region from a dry vapour layer close to the steel plate.The thickness of this vapour layer is determined by the solution of a generalised Stefan problem. The second process concerns the macroscopic modelling of pig iron production in blast furnaces. In the simplest scenario, the blast furnace may be roughly divided into a porous solid region. overlaying a hot high pressure gaseous zone. The gas reacts with the solid in a thin "intermediate region" at the base of the solid region and it is in this intermediate region that the pig iron is produced. A free boundary model is proposed for the location of the intermediate region and its stability is investigated.     

Free Boundary Problems in the Steel Industry

This paper considers two novel free boundary problems that emerge from modelling processes basic to steel manufacture. The first process concerns the spray cooling of hot steel sheet during the process of continuous casting. Here, an important practical consideration is the non-monotonicity of the measured heat transfer from the steel as a function of the steel temperature. In order to understand this phenomenon, a two-phase flow model is written down for the heating and vapourisation of the water spray. This model relies on a microscale analysis of droplet vapourisation and, in a steady state, it reduces to a coupled system of nonlinear ordinary differential equations for the spray temperature and water content. This system predicts the conditions for the existence or otherwise of a free boundary separating the two-phase region from a dry vapour layer close to the steel plate.The thickness of this vapour layer is determined by the solution of a generalised Stefan problem. The second process concerns the macroscopic modelling of pig .iron production in blast furnaces. In the simplest scenario, the blast furnace may be roughly divided into a porous solid region overlaying a hot high pressure gaseous zone. The gas reacts with the solid in a thin “intermediate region“ at the base of the solid region and it is in this intermediate region that the pig iron is produced. A free boundary model is proposed for the location of the intermediate region and its stability is investigated.     

Ultrasonic wave propagation loss factor in composite in terms of constituent properties

A model is developed to relate the energy loss factors of ultrasonic longitudinal waves propagating in the principal directions of a unidirectional graphite fiber composite to the composite constituent properties. All the constituents are assumed to behave as linear viscoelastic materials with energy dissipation properties defined by loss factors. It is found that by introducing a new constituent called the interface material, the composite and constituent properties can be brought into consistency with simple series and parallel models. An expression relating the composite loss factors to the loss factors of the constituents is derived and its coefficients are evaluated.     

Resin transfer moulding: mathematical modelling and numerical simulations

This study accounts for the necessity of developing a proper mathematical model of composite material processes in order to implement it in a numerical simulation. The process considered here is the resin transfer moulding (RTM). It essentially consists of the injection of a polymeric resin in a porous pre-form of reinforcing elements. Process simulation is of considerable value in assessing production parameters and in improving the quality of manufactured products. The proposed model is deduced in the framework of mixture theory. In particular, a solid–liquid–air mixture is considered. The porous solid is deformable and its mechanical behavior is non-linear elastic. The model consists of a set of partial differential equations, time dependent, defining two coupled problems: mechanical equilibrium and diffusion in a permeable medium. The equations are written, at first, in the Eulerian formulation and then, considering a set of material coordinates fixed on the porous solid, in the Lagrangian formulation. The mathematical problem, generated by the model, is defined in the whole domain occupied by the mixture of three-phases. It is shown that, if the transitional layer between the region wet by the infiltrating liquid and the one not reached by the liquid is thin, then the formulation proposed is equivalent to the classical formulation where different sets of equations are used for different regions. The problem is solved numerically by means of finite element method. The simulations developed use the ABAQUS FEA package. Two 3D infiltration problems are simulated. Although the simulations do not refer to any practical applications, the results show the usefulness of the model in the assessment of process parameters in true industrial problems.     

Thermo-plasticity of metal foams

The thermo-mechanical behaviour of a fluid-filled metal foam is influenced by the characteristics of the solid skeleton and the pore-fluid. During the deformation process both constituents exchange momentum and energy. Based on the theory of porous media a fluid-filled foam is modeled as a binary mixture consisting of a metal skeleton and a pore-gas. The porous solid skeleton is assumed to behave thermo-elasto-plastically. The pore-gas is considered as an ideal gas. For both constituents different phase temperatures and thermo-mechanical coupling mechanisms are taken into account. The resulting system of differential equations is solved using the finite element method. An example demonstrates the applicability and validity of this approach.     

Diffusion controlled smoulder propagation in a thin slab

Complex variable techniques are used to determine the shape of the smouldering reaction front and the concentration of the oxidizer behind the front for steady smoulder propagation in a solid slab of exothermically reacting material. It extends an earlier free boundary problem of Adler and Herbert which considered diffusion controlled smoulder propagation in a half-space. The region behind the reaction front is assumed to be porous, the oxidizer diffusing from both planar surfaces to the front, where its concentration vanishes. Suitable scaling allows the oxidizer concentration to be expanded in powers of a small parameter. The resulting coupled differential equations for the coefficients are solved in terms of functional equations. Some consideration is given to the regions where the front meets the planar surfaces. It is shown that, close to the leading edge, the surface concentration varies monotonically with distance from the edge.     

Bifurcation of a solid circular elastic cylinder under finite extension and torsion

Summary The problem of bifurcation of a solid circular cylinder subjected to finite extension and torsion is investigated using the theory of small deformations superposed on large elastic deformations. The material of the cylinder is assumed to be isotropic, elastic, homogeneous and incompressible. A numerical scheme is adopted to solve the system of partial differential equations and the associated boundary conditions governing the problem for a class of strain-energy functions. The numerical results obtained determined the critical twist corresponding to a given extension of the cylinder.     

Elastic green's function for a bimaterial composite solid containing a crack inclined to the interface

An analytical closed-form expression is derived for the elastic Green's function of a bimaterial composite solid containing a planar interface and a straight crack inclined at an arbitrary angle with the interface. The crack tip is assumed to be at the interface. Both the constituent materials of the composite are assumed to anisotropic. The Green's function satisfies the interfacial boundary conditions of continuous tractions and displacements, and zero tractions at the crack surfaces. The boundary conditions are satisfied by using the virtual force technique. The determination of the virtual forces requires solutions of a Hilbert problem which is obtained by using an orthogonal complex transform. The method is illustrated by applying it to a copper/nickel composite. The Green's function should be useful in the boundary-element method of calculating the stress and the displacement field in the solid.     

A two-phase model for dry density-varying granular flows

A granular flow is normally comprised of a mixture of grain-particles (such as sand, gravel or rocks) of different sizes. In this study, dry granular flows are modeled utilizing a set of equations akin to a two-phase mixture system, in which the interstitial fluid is air. The resultant system of equations for a two-dimensional configuration includes two continuity and two momentum balance equations for the two respective constituents. The density variation is described considering the phenomenon of air entrainment/extrusion at the flow surface, where the entrainment rate is assumed to be dependent on the divergent or convergent behavior of the solid constituent. The density difference between the two constituents is extremely large, so, as a consequence scaling analysis reveals that the flow behavior is dominated by the solid species, yielding small relative velocities between the two constituents. A non-oscillatory central (NOC) scheme with total variation diminishing (TVD) limiters is implemented. Three numerical examples are investigated: the first being related to the flow behaviors on a horizontal plane with an unstable initial condition; the second example is devoted to simulating a dam-break problem with respect to different initial conditions; and in the third one investigates the behavior of a finite mass of granular material flowing down an inclined plane. The key features and the capability of the equations to model the behavior are illustrated in these numerical examples.     

Magnetothermoelastic interaction in an infinite elastic continuum with a cylindrical hole subjected to ramp-type heating

A solution is given for the induced temperature and stress fields in an infinite transversely isotropic solid continuum with a cylindrical hole. The solid medium is considered to be exposed to a magnetic field and the cavity surface is assumed to be subjected to a ramp-type heating. Green and Lindsay model has been used to account for finite velocity of heat conduction. Magnetic field induced within the cavity is also determined. The problem is solved analytically by using integral transform technique.     

Failure analysis of fiber-reinforced composite laminates subjected to biaxial loads

A nonlinear constitutive model for a single lamina is proposed for the failure analysis of composite laminates. In the material model, both fiber and matrix are assumed to behave as elastic-plastic and the in-plane shear is assumed to behave nonlinearly with a variable shear parameter. The damage onset for individual lamina is detected by a mixed failure criterion, composed of the Tsai-Wu criterion and the maximum stress criterion. After damage takes place within the lamina, the fiber and in-plane shear are assumed to exhibit brittle behavior, and the matrix is assumed to exhibit degrading behavior. The proposed nonlinear constitutive model is tested against experimental data and good agreement is obtained. Then, numerical analyses are carried out to study the failure behavior of symmetric angle-ply composite laminates and symmetric cross-ply composite laminates subjected to biaxial loads. Finally, the conclusions obtained from the numerical analysis are given.     

A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures

A continuum theory of a multiphase mixture is formulated. In the basic balance laws we introduce an additional balance of equilibrated forces to describe the microstructural response according to Goodman &; Cowin [11] and Passman et al. [23] for each constituent. Based on the Müler-Liu form of the second law of thermodynamics a set of constitutive equations for a viscous solid-fluid mixture with microstructure is derived. These relatively general equations are then reduced to a system of ordinary differential equations describing a steady flow of the solid-fluid mixture between two horizontal plates. The resulting boundary value problem is solved numerically and results are presented for various values of parameters and boundary conditions. It is shown that simple shearing generally does not occur. Typically, for the solid phase, in the vicinity of a boundary, if the solid-volume fraction is low, a layer of high shear rate occurs, whose thickness is nearly between 5 and 15 grain diameters, while if the solid-volume fraction is high, an interlock phenomenon occurs. The fluid velocity depends largely on the drag force between the constituents. If the drag coefficient is sufficiently large, the fluid flow is nearly the same as that of the solid, while for a small drag coefficient, the fluid shearing flow largely decouples from that of the solid in the entire flow region. Apart from this, there is a tendency for solid particles to accumulate in regions of low shear rate.     

Fluid-fluid and -solid interaction problems: Variational principles revisited

We systematically review some unified variational principles for a strong interaction problem in both a stratified fluid region and a fluid-solid region. The problem is described by a general Lagrangian formulation for an anisotropic elastic solid region, either surrounded by an incompressible non-Newtonian fluid region or surrounding the fluid region. In the first part, we express the fundamental equations of the regular fluid and solid regions in differential form. Then, we deduce the variational principles respectively from the principle of virtual power and the principle of virtual work for the fluid and solid regions. The physics principles are modified through an involutory transformation together with a dislocation potential. In the second part, we similarly establish some multi-field variational principles for a stratified fluid of two or more distinct fluid layers of different thicknesses and mass densities. In the third part, we derive the variational principles for the interior and exterior interaction problems in a fluid region with a surface piercing solid, within either a rigid or an elastic structure. The variational principles, which operate on all the field variables lead to the fundamental equations of the regions, including the interface conditions, as their Euler-Lagrange equations. Some special cases of the variational principles are given.     

A reciprocal theorem for a mixture of interacting continuous media

Using a linearized theory of interacting continuous media composed of linear elastic solid and linear viscous fluid, a reciprocal theorem is derived which relates the solution of one initial-boundary value problem to another. As an illustration of the reciprocal theorem, we obtain the early time displacement and velocity field of a mixture occupying an infinite region and subjected to an impulsively applied moving point load acting on the solid constituent.     

Inflation of hyperelastic cylindrical membranes as applied to blow moulding. Part I. Axisymmetric case

The free and confined axisymmetric inflation of a cylindrical membrane is considered in this study. The problem is formulated to include single and multiple contacts between the cylinder and the surrounding solid boundary. The material is assumed to obey the Mooney–Rivlin constitutive equation. Unlike various finite element based discretization approaches, the present formulation is based on a set of ordinary differential equations governing the deformation field. The resulting two-point-boundary-value problem is solved using the finite element method. Comparison between theory and experiment leads to good agreement. Stick condition upon contact is assumed. Examples of complex geometries including a blow-moulded bottle are treated. An example of simultaneous stretch and inflation is also given.     

A NUMERICAL STUDY OF THE FLOW OF GRANULAR MATERIALS BETWEEN TWO VERTICAL FLAT PLATES WHICH ARE AT DIFFERENT TEMPERATURES

The present study considers an assembly of spherical particles, densely packed, between two vertical flat plates which are at different temperatures. A continuum model is used and the flow due to such a temperature difference is investigated. For a fully developed flow of these materials, the governing equations reduce to a system of coupled, non-linear ordinary differential equations. The equations are integrated numerically. In so doing, the equations are discretized using the conventional central finite difference approximation technique and the unknown conditions are assumed to reduce the problem to an initial value problem. After the trial solutions the known boundary conditions at the edge of the integration interval are compared with the correspondent values provided by the trial solutions. If the solutions at this point do not agree with the known boundary conditions the Newton-Raphson method is used to correct the initial guesses and the iteration is repeated. This procedure continues until the solutions converge to the given values. The technique is found to be powerful for this type of application.     

A NUMERICAL STUDY OF THE FLOW OF GRANULAR MATERIALS BETWEEN TWO VERTICAL FLAT PLATES WHICH ARE AT DIFFERENT TEMPERATURES

ABSTRACT

The present study considers an assembly of spherical particles, densely packed, between two vertical flat plates which are at different temperatures. A continuum model is used and the flow due to such a temperature difference is investigated. For a fully developed flow of these materials, the governing equations reduce to a system of coupled, non-linear ordinary differential equations. The equations are integrated numerically. In so doing, the equations are discretized using the conventional central finite difference approximation technique and the unknown conditions are assumed to reduce the problem to an initial value problem. After the trial solutions the known boundary conditions at the edge of the integration interval are compared with the correspondent values provided by the trial solutions. If the solutions at this point do not agree with the known boundary conditions the Newton-Raphson method is used to correct the initial guesses and the iteration is repeated. This procedure continues until the solutions converge to the given values. The technique is found to be powerful for this type of application.     

Cylindrical cavity expansion and contraction in pressure sensitive geomaterials

Summary A broad class of plane-strain axially-symmetric deformation patterns in geomaterials is studied within the framework of large strain pressure-sensitive plasticity. Invariant, non-associated deformationtype theories are formulated for the Mohr-Coulomb (M-C) and Drucker-Prager (D-P) solids with arbitrary hardening and accounting for an initial hydrostatic state of stress. With the M-C model we arrive at a single first order differential equation, while for the D-P solid an algebraic constraint supplements the governing differential equation. The analysis centers on the effective stress as the independent variable. A simplified treatment is given for the cavitation limit and some useful relations are derived for thin walled cylinders. The theory is applied to the triaxial calibration test for Castlegate sandstone and then used to simulate the hole closure problem. Numerical examples are provided for the case of a cavity embedded in an infinite medium subjected to external or internal pressure. Results for the D-P inner cone model were found to be in close agreement with those obtained from the M-C model.     

Thermostatics of an infinite mixture of two elastic solids and a spherical thermal inclusion problem

This work is concerned with the linear theory of a binary mixture of two elastic solids. With the help of displacement potentials, two differential equations that govern the displacement of each constituent are obtained. Then, more generalised forms of Betti's reciprocal theorem and Maysel's formula for the mixture are found. Finally, a solution of spherical thermal inclusion problem in an infinite mixture of two elastic solids is obtained by using generalised Maysel's formula and by direct integration of the governing differential equations.     

Formulation and solution of problems of tool positioning on a single machining centre

We extend previous work on the construction of a computer technique used to formulate and solve a tool positioning problem concerning the efficient use of a single machining centre. The problem is formulated as an integer programming model. The problem is to identify which pockets of a circular tool magazine should be used to accommodate specific tools, during machine setup, so as to minimize the makespan of a given sequence of operations. We now analyse the more realistic situation in which it is assumed that the given sequence of operations is to be carried out repeatedly. Consequently, the pockets of the magazine must contain the same classes of tools at the end of each cycle of operations as they did at the beginning, so that the process can be repeated in an identical fashion. The extended problem is also formulated as an integer programming model, capable of being solved using commercially available software. Problems of a realistic size can then be used to set up operating conditions for a machining centre