The limit equilibrium of an anisotropic medium under the general plasticity condition

A theory of the limit equilibrium of an anisotropic medium under the general plasticity condition in the plane strain state is developed. The proposed yield criterion (the limit equilibrium condition) is obtained by combining the von Mises–Hill yield criterion of an ideally plastic anisotropic material and Prandtl's limit equilibrium condition for a medium under the general plasticity law. It is shown that the problem is statically determinate, i.e., if the boundary conditions are specified in stresses, the stress state in plastic region can only be obtained using equilibrium equations. It is established that the equations describing the stress state are hyperbolic and have two families of characteristic curves that intersect at variable angles. In deriving the equations describing the velocity field, the material is assumed to be rigid plastic, and the associated law of flow is applied. It is shown that the equations for the velocities are also hyperbolic, and their characteristic curves are identical with those of the equations for stresses. However, the directions of the principal values of the stress and strain rate tensors are different due to the anisotropy of the material. The characteristic directions differ from the isotropic case in that the normal and tangential components of the stress tensor do not satisfy the limit conditions. It is established that the equations obtained allow of partial solutions, and in this case, at least one family of characteristic curves consists of straight lines. The conditions along the lines of discontinuity of the velocity are investigated, and it is shown that, as in the isotropic case, these are characteristic curves of the system of governing equations. In the anisotropic formulation, the well-known Rankine problem of the limit state of a ponderable layer is solved. From an analysis of the velocity field it is shown that plastic flow of the entire layer is possible only for a slope angle equal to the angle of internal friction. For slope angles less than the angle of internal friction, the solutions obtained are solutions of problems of the pressure of the medium on the retaining walls. The change in this pressure as a function of the parameters of anisotropy is investigated, and turns out to be significant.     

The indentation of a flat punch into an ideal rigid-plastic half-space under the action of shear contact stresses

A general plane problem of the impression of a flat punch into a rigid-plastic half-space under the action of transverse and longitudinal shear contact stresses is considered. The condition of complete plasticity and the hyperbolic equations of the general plane problem of the theory of ideal plasticity [1] are used. The reduction of the limit pressure on the punch is determined as a function of the shear contact stresses.     

Plastic flow in a conical channel: Qualitative features of the solutions under different yield conditions

The problem of the convergence of the solutions of problems of plasticity theory, with a yield condition which depends on the hydrostatic stress, to solutions based on classical plasticity theory with von Mises or Tresea conditions is considered, with a particular choice of the parameters of the material model. For the case of axisymmetric flow of material in a channel with converging and diverging walls, solutions according to two plasticity theories with a yield condition which depends on the hydrostatic stress are compared with the classical solution. It is shown that only the solution using Spencer's model shows all the main features of the classical solution. As the internal criterion of the choice of the preferred plasticity theory when examining a special class of problems, it is suggested that the criterion of the convergence of the solutions to the solutions of classical plasticity theory should be used.     

Finite element modelling of surface waves in a channel

A variational formulation of the vertically-integrated differential equations for free surface wave motion is presented. A finite element model is derived for solving this nonlinear system of hydrodynamic equations. The time integration scheme employed is discussed and the results obtained demonstrate its good stability and accuracy.Several applications of the model are considered: the first problem is an open channel of uniform depth and the second an open channel of linearly varying depth. The ‘inflow’ boundary condition is prescribed in terms of the velocity which represents a wavemaker and/or a flow source, while the ‘outflow’ boundary condition is specified in terms of the water elevation. The outflow condition is adjusted for two cases, a reflecting boundary (finite channel) and a non-reflecting boundary (open-ended channel). The latter boundary condition is examined in some detail and the results obtained show that the numerical model can produce the non-reflecting boundary that is similar to the analytical radiation condition for waves. Computational results for a third problem, involving wave reflection from a submerged cylinder, are also presented and compared with both experimental data and analytical predictions.The simplicity and the performance of the computational model suggest that free surface waves can be simulated without excessively complicated numerical schemes. The ability of the model to simulate outflow boundary conditions properly is of special importance since these conditions present serious problems for many numerical algorithms.     

Homogenization of boundary layer of pseudo-plastic fluid in the presence of rapidly oscillating external forces

In this paper, we study the homogenization problem for equations of magnetohydrodynamic boundary layer of pseudo-plastic fluid. It is assumed that the external flow velocity and the external magnetic field are described by oscillating functions and the frequency depends on a small parameter. In von Mises variables and in Cartesian variables, we construct the homogenized problem, establish strong convergence of solutions in a special norm, and estimate the rate of this convergence. We show that in von Mises variables the convergence rates in different norms are of different orders of smallness.     

Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type

We study the solvability of a boundary value problem for a system of nonlinear second-order partial differential equations under given boundary conditions, which describes the equilibrium of elastic shallow shells with hinged edges in the framework of the Timoshenko shear model. The study method implies the reduction of the original system of equations to a single nonlinear differential equation whose solvability is proved with the use of the contraction mapping principle.     

Classical solutions to phase transition problems are smooth

We prove that classical C¹–solutions to phase transition problems, which include the two–phase Stefan problem, are smooth. The problem is reduced to a fixed domain using von Mises variables. The estimates are obtained by frozen coefficients and new L_p estimates for linear parabolic equations with dynamic boundary condition. Crucial ingredients are the observation that a certain function is a Fourier multiplier, an approximation procedure of norms in Besov spaces and Meyer' approach to Nemytakij operators.     

A hydrodynamic interpretation of a problem in the theory of the dimensional electrochemical machining of metals

A method of determining the form of the anode-blank boundary for a specified form of the cathode tool in plane problems of the theory of the dimensional electrochemical machining of metals is proposed. Within the assumptions made, the anode-blank boundary is divided into a working zone, in which solution of the metal occurs, and a region next to it in which the machining ceases. The initial problem is reduced to a problem of plane-parallel potential flow of an ideal liquid with non-linear conditions on its surface. The point which separates these two regions of the anode boundary corresponds to the point where the jet separates from the solid boundary. The Brillouin-Villat smooth separation condition is specified when compiling the closed system of equations at the point where the jet separates.     

Wave Propagation in an Elastic Medium Intersected by Systems of Parallel Fractures

An elastic anisotropic medium intersected by systems of parallel fractures is investigated. Every fracture is considered as a plane boundary with jumps of displacements and stresses, and these jumps are linear functions of displacements and stresses averaged on the boundary. For this medium, an effective model is constructed by the method of matrix averaging. The equations of this model describe wave propagation in the given medium and are more complicated than the equations of elasticity theory. In particular cases, the equations obtained are converted to the equations of elastic media. On the basis of the equations of the effective model, expressions for the densities of the kinetic and potential energies are derived, and conditions of absoption in the medium are established. Bibliography: 15 titles.     

Thermal stress analysis of current-carrying media containing an inclusion with arbitrarily-given shape

The presence of inclusions in metal-based composites subjected to an electric current or a heat flux induces thermal stresses. Inclusion geometry is one of the important parameters in the stress distribution. In this study, the plane problem of an arbitrarily-shaped inclusion embedded in an infinite conductive medium is investigated based on the complex variable method. The shape of the inclusion is defined approximately by a polynomial conformal mapping function. Faber series and Fourier expansion techniques are used to solve the corresponding boundary value problems. The obtained results show that the shape, bluntness and rotation angle of the inclusion have a significant effect on the stress concentration around the inclusion induced by the far-field electric current. In addition, for the considered inclusion-matrix system under given electric loading, a lower amount of the Von Mises stress concentration than that around a circular inclusion could be achieved by appropriate selection of the inclusion shape and orientation.     

Superposition of the Nadai and prandtl solutions for the system of two-dimensional ideal plasticity

A general algorithm is presented for transforming the exact solutions of the system of plane ideal plasticity of the Mises medium by using the superposition principle for solutions which arises as a corollary of the fact that the original system admits an infinite-dimensional symmetry group. As an example, there is considered the relation between the known exact solutions: the Prandtl solution for a thin layer compressed by rough solid plates and the Nadai solution for the radial distribution of stresses in a convergent channel in the shape of a flat wedge.     

Layerwise sensing in X-ray tomography in the polychromatic case

An X-ray tomography problem that is an inverse problem for the transport differential equation is studied. The absorption and single scattering of particles are taken into account. The suggested statement of the problem corresponds to stepwise and layerwise sensing of an unknown medium with initial data specified as the integrals of the outgoing flux density with respect to energy. The sought object is a set on which the coefficients of the equations suffer a discontinuity, which corresponds to searching for the boundaries between the different substances composing the sensed medium. A uniqueness theorem is proven under rather general assumptions and a condition guaranteeing the existence of the sought lines. The proof is constructive and can be used for developing a numerical algorithm.     

A nonlocal problem for singular linear systems of ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite half-interval. This system is supplemented by the boundedness condition for solutions and a nonlocal linear condition specified by the Stieltjes integral. A method for approximating the resulting problem by a problem posed on a finite interval is proposed, and the properties of the latter are investigated. A numerically stable method for solving this problem is examined. This method uses an auxiliary boundary value problem with separated boundary conditions.     

Mathematical simulation of acoustic wave scattering in fractured media

The simulation of acoustic waves in fractured media is considered. A self-consistent field model is proposed that describes the formation of a scattered field and the attenuation of the incident field. For the total field, a wave equation with a complex velocity is derived and the corresponding dispersion equation is studied. A frequency-dependent field damping law and an energy variation law are established. An initial and a boundary value problem for waves in a fractured medium is addressed. A finite-difference scheme for the initial value problem is constructed, and a condition for its stability is established. Numerical results are presented.     

Hyperbolic variational inequalities in problems of the dynamics of elastoplastic bodies

Results of a study of variational inequalities appearing in dynamic problems of the theory of elastic-ideally plastic Prandtl-Reuss flow are given. The concept of a generalized solution is formulated for the general-type inequality and is used to construct the complete system of relations for a strong discontinuity. A priori estimates are obtained which make it possible to prove the uniqueness and continuous dependence “in the small” on time of the solutions of the Cauchy problem and initial-boundary value problems with dissipative boundary conditions, as well as the estimates of the nearness of the solutions of the variational inequality and of the system of equations with a small parameter describing the elasto-viscoplastic deformation of the bodies. The problem of the propagation of plane waves in an elastoplastic half-space with initial stresses is used as an example to illustrate the difference between the discontinuous solutions with the Mises yield condition and with the Tresca-St Venant consition in the theory of flows.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

理想塑性轴对称问题的一般方程

本文引用速度势函数,将理想塑性轴对称问题化为二个非线性的偏微分方程,根据导得的方程讨论了Haar-Kármán假设对于Mises屈服准则及与其相关联的流动法则的协调性问题。     

Asymptotic determination of the formation process of nonlinear distortion of one-dimensional pulses in a layered medium : PMM vol.40, n 6, 1976, pp. 1093–1103

Nonlinear effects in the propagation, reflection, and refraction of one-dimensional pulses in a medium consisting of two layers lying on a half-space are considered and analyzed. Properties of layers and of the half-space are different, and stresses are defined by an expansion in powers of strains. The initial pulse of finite duration is specified in the form of boundary condition at the surface of the external layer either for the deformation or for the dislocation rate, and the problem of wave pattern when the initial pulse amplitude tends to zero,i.e. in the case of small nonlinear effects, is solved.Problem is solved by the method of successive integration of nonhomogeneous linear wave equations, in which the solution of the linear problem is taken as the first approximation and the subsequent approximations are derived by approximating the nonlinear terms with the use of the preceding approximation.     

AN ARTIFICIAL BOUNDARY CONDITION FOR THE INCOMPRESSIBLE VISCOUS FLOWS IN A NO-SLIP CHANNEL

1.IntroductionWhencomputingthenumericals0luti0nsofviscousfluidfl0wproblemsinallun-boundedd0main,0neoftenintroducesartificialboundaries,andsetsupanartificialbopundarycondition0nthem;thenthe0riginalproblemisreducedtoaproblemonab0undedc0mputationald0main.InordertoIimitthecomputatio11alcost,theseboundariesmustnotbet00farfromthedomainofinterest.Theref0re,theartificialboundaryc0nditi0nsmustbegoodapprotimationt0the"exact"boundaryconditions(sothatthes0lutionoftheproblemintheboundeddonlainisequaltothes…     

Analysis of Time–Domain Maxwell's Equations for 3-D Cavities

Time–domain Maxwell's equations are studied for the electromagnetic scattering of plane waves from an arbitrarily shaped cavity filled with nonhomogeneous medium. A transparent boundary condition is introduced to reduce the problem to the bounded cavity. Existence and uniqueness of the model problem are established by a variational approach and the Hodge decomposition. The analysis forms a basis for numerical solution of the model problem.