Iteration Algorithms for Solving Boundary-Value Problems of the Theory of Small Elastic-Plastic Strains on the Basis of the Mixed Finite Element Method

A mixed projection-mesh scheme has been formulated for the solution of nonlinear boundary-value problems of the theory of small elastic-plastic strains in displacements, strains, and stresses. Iteration algorithms of the solution of nonlinear equations of the mixed method have been considered. Numerical results of the solution of two model problems are presented. The data obtained on the basis of the classical and mixed finite element methods are compared.__________Translated from Problemy Prochnosti, No. 3, pp. 111 – 127, May – June, 2005.     

Mixed Projection-Mesh Scheme of the Finite-Element Method for the Solution of Problems of the Elasticity Theory

A mixed projection-mesh scheme for solving the boundary-value problems of the elasticity theory has been formulated. Correctness and convergence of the mixed approximations for strains and displacements are studied. Application of the numerical integration has been analyzed, and the obtained results are given. The theory of generalized functions and the functional analysis methods have been used for the convergence and accuracy estimates. Iteration algorithms for solving the discrete problems have been proposed.     

Mixed projection-mesh scheme of the finite-element method to solve boundary-value problems describing the non-isothermal processes of elastoplastic deformation

A mixed projection-mesh scheme for solving a boundary-value problem of thermal plasticity is formulated in a quasi-static statement when the process of non-isothermal elastoplastic deformation of a body is a sequence of equilibrium states. In this case, the stress-strain state depends on the loading history, and the process of inelastic deformation is to be observed over the whole time interval under study. The correctness and convergence of the mixed approximations for stresses, strains and displacements are investigated as applied to the solution of nonlinear boundary-value problems that describe the non-isothermal processes of active loading taking into account the initial strains dependent on the history of deformation and heating. The properties of the projecting operators are studied in detail, and on this basis, the condition that ensures the existence, uniqueness and stability of solution is formulated. The results of the analysis of special formulas of the interpolation-type numerical integration are presented, the use of which considerably simplifies the computation procedure for solving equations of the mixed method. The convergence and accuracy estimations are based on the results of the theory of the generalized boundary-value problems and methods of the functional analysis. According to the estimations obtained, the accuracy of solution of a finite-dimensional problem at the initial stages of loading should be sufficient to avoid the effect of increase of the first coefficients in the expansion of the total error on the accuracy of solution of the elastoplastic problem at the subsequent stages of loading. __________ Translated from Problemy Prochnosti, No. 3, pp. 87–117, May–June, 2007.     

Application of the mixed approximation to the solution of two-dimensional problems of the theory of small elastoplastic strains using the finite element method

A triangular finite element that provides the stability and convergence of the mixed approximation is used for the solution of two-dimensional boundary-value problems of the theory of small elastoplastic strains. A system of resolving matrix equations of a mixed type is presented for the solution of which a three-layer iteration algorithm with a preconditioning matrix is used. Numerical results for the solution of model problems obtained by the classical and combined finite element methods are compared. __________ Translated from Problemy Prochnosti, No. 2, pp. 124–136, March–April, 2006.     

Mixed Approximation Scheme of the Finite-Element Method for the Solution of Two-Dimensional Problems of the Elasticity Theory

For the solution of two-dimensional boundary-value problems of the elasticity theory, a triangular finite element, ensuring stability and convergence of mixed approximation, is proposed. The system of resolving equations of the mixed method is derived with account of strict satisfaction of static boundary conditions at the surface. To solve matrix equations of the mixed method, various algorithms of the conjugate-gradient method with the pre-conditional matrix have been considered. Numerical data on convergence and accuracy of the solution for a number of test problems of the elasticity theory and fracture mechanics are given. The results obtained by the conventional and mixed finite-element method approaches are compared.     

Three-dimensional problem of the theory of elasticity in strains

A three-dimensional problem of the theory of elasticity is formulated in strains. The constitutive equations are presented and the subject of obtaining the general solutions of these equations is discussed. It is shown that the six Saint-Venant strain compatibility equations are not independent of one another. These equations are divided into two groups. The three equations of the second group can be obtained from the three equations of the first group, and vice versa.Translated from Problemy Prochnosti, Nos. 5–6, pp. 69–73, May–June, 1995.     

Growth of plasticity bands around the tip of an arbitrarily oriented crack in a semiinfinite thin plate

Summary Elastoplastic equilibrium is considered for a thin semiinfinite plate containing an arbitrarily oriented rectilinear edge crack. It is assumed that the plastic strains are localized along a narrow band arising from the crack tip. The elastoplastic treatment is then reduced to a planar treatment in the theory of elasticity with an edge kinked crack whose edges are subject to defined forces. The solution is derived by means of singular integral equations. Numerical results have been obtained for a tensile load acting at infinity for a constant pressure acting on the crack edges. An estimate is made of how the free edge of the plate affects the magnitude and orientation of the plasticity band.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 3, pp. 25–31, May–June, 1992.     

Stress analysis of axisymmetric shear deformable cross-ply laminated circular cylindrical shells

A generalized mixed theory for bending analysis of axisymmetric shear deformable laminated circular cylindrical shells is presented. The classical, first-order and higher-order shell theories have been used in the analysis. The Maupertuis–Lagrange (M–L) mixed variational formula is utilized to formulate the governing equations of circular cylindrical shells laminated by orthotropic layers. Analytical solutions are presented for symmetric and antisymmetric laminated circular cylindrical shells under sinusoidal loads and subjected to arbitrary boundary conditions. Numerical results of the higher-order theory for deflections and stresses of cross-ply laminated circular cylindrical shells are compared with those obtained by means of the classical and first-order shell theories. The effects, due to shear deformation, lamination schemes, loadings ratio, boundary conditions and orthotropy ratio on the deflections and stresses are investigated.     

Consideration of some dilute-solution phenomena based on an expression for the Gibbs free energy

Rigorous expressions based on the Lennard-Jones (6–12) potential, are presented for the Gibbs and Helmholtz free energy of a dilute mixture. These expressions give the free energy of the mixture in terms of the thermodynamic properties of the pure solvent, thereby providing a convenient means of correlating dilute mixture behavior with that of the pure solvent. Expressions for the following dilute binary solution properties are derived: Henry's constant, limiting activity coefficients with their derivatives, solid solubilities in supercritical gases, and mixed second virial coefficients. The Henry's constant expression suggests a linear temperature dependence; application to solubility data for various gases in methane and water shows a good agreement between theory and experiment. In the thermodynamic modeling of supercritical fluid extraction, we have demonstrated how to predict new solubility-pressure isotherms from a given isotherm, with encouraging results. The mixed second virial coefficient expression has also been applied to experimental data; the agreement with theory is good.Paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Mathematical Model of Nonisothermal Turbulent Flow of Gas Suspension in a Pipe on the Basis of the Mixed Eulerian–Lagrangian Representation

A mathematical model of nonisothermal turbulent flow of a gas suspension in a pipe is formulated on the basis of the mixed Euler–Lagrange representation. The testing of the model revealed good agreement of the calculation results with experimental data and with the data obtained using the continuum model developed by us previously. The use of two essentially different (continuum and mixed) approaches to the simulation of dusty flows leads to nearly identical results for the fields of averaged velocities and temperatures and for characteristics such as the coefficient of accommodation of the tangential component of the velocity and the intensity of the particle collisions with the wall surface.     

Numerical modeling of a plane hydrodynamic problem for a viscoelastic lubricant. The case of short relaxation times

Numerical solutions of a two-dimensional isothermal problem of flow of a relaxing fluid between two rotating cylindrical surfaces have been constructed. All the results have been obtained by solution of a complete system of hydrodynamic equations. The influence of the internal relaxation time on the carrying capacity of a lubricating film and the flow structure has been analyzed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 77, No. 6, pp. 162–167, November–December, 2004.     

Modelling of the thermodynamic properties of the water–ammonia mixture by three different approaches

This paper deals with the modelling of the thermodynamic properties of the water–ammonia refrigerant mixture. Three different approaches are formulated and compared. The first is an empirical approach based on a free enthalpy model of the mixture considered as the resultant of the properties of its pure components and of an excess term corresponding to the deviation to the ideal solution concept. Secondly, a semi-empirical approach based on the PATEL and TEJA cubic equation of state is considered. Finally, a theoretical approach formulated as PC-SAFT (perturbed chain statistical associating fluid theory) equation of state is treated. A comparison of these three methods proves the superiority of PC-SAFT in predicting and extrapolating the thermodynamic properties of the water–ammonia system up to very high temperatures and pressures.     

A variant of the endochronic theory of plasticity with a singular kernel for describing complex deformation processes

The effectiveness of the application of equations of state with a weak peculiarity of the endochronic theory of plasticity in describing complex processes is investigated. The influence exerted by the magnitude of the bend angle of the stress path and various plastic-strain paths on prediction of the material's mechanical behavior is analyzed. The results obtained are compared with the solution for a model employing a regular kernel.Translated from Problemy Prochnosti, No. 4, pp. 97–102, April, 1990.     

Investigation of the Creep of Plates of Complex Shape by the Method of <Emphasis Type="Italic">R</Emphasis>-Functions

By using the Lagrange variational principle, the theory of R-functions, and the Ritz method in combination with the Runge-Kutta-Merson method for the integration of initial-value problems with respect to time, we developed a method for the numerical analysis of creep and long-term strength of laminated structural elements. The reliability of the proposed structural relations for the solution of boundary-value problems and the developed software is substantiated by the practical convergence of the obtained results with the increase in the number of coordinate functions, the increase in the accuracy of calculation of elements of the Ritz matrix and the accuracy of integration with respect to time, and good agreement of these results with the available numerical data. We study the processes of creep and damage for plates subjected to bending depending on the conditions of fixing and the shape of the plate.__________Translated from Problemy Prochnosti, No. 3, pp. 104 – 110, May – June, 2005.     

Thermal conductivity of mixtures with interpenetrating components

The basic principles of investigation of transport processes used in generalized conduction theory are formulated. Studies in which a structure with interpenetrating components has been used for determining electrical and thermal conductivity are analyzed. It is shown that using models of such structures it is possible to calculate the thermal conductivity of gas mixtures, liquid solutions, solid disperse systems, certain alloys, and granular and fibrous materials.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 19, No. 3, pp. 562–577, September, 1970.     

Stress singularity in a substrate-coating system

A problem of the theory of elasticity is solved for a substrate—coating system with the coating free edge of various configurations. A biharmonic equation of the theory of elasticity in the polar coordinates is considered, for the solution of which the Airy function is proposed to be used. The dependences of the singularity on the ratio of the elastic characteristics of the substrate—coating system and on the configuration of the coating free edge have been obtained. The order of the stress singularity has been found for plasma-sprayed coatings. In particular cases, the problem has been shown to degenerate into the fracture mechanics problem on a crack in a homogeneous material or into the problem of the theory of elasticity for homogeneous wedges and notches.Translated from Problemy Prochnosti, No. 6, pp. 119–127, November–December, 2004     

Analysis of the Efficiency in Certain Mass‐Exchange Models with Different Equilibrium Dependences

The efficiency of mass exchange in the complex and hypothetical models and in the Murphree and Hausen models for concurrent and cross motions of the vapor and liquid phases has been compared with the use of equilibrium dependences of the y = mx and y = ax + b type. Industrial data on separation of furfurol–water and ethanol–water binary mixtures have been employed in the calculations. Satisfactory convergence of the results has been obtained.     

Stress-intensity coefficients around the vertices of cracks and rigid inclusions in strips with clamped or free boundaries

Summary Singular integral equations have been applied to the state of stress and strain in an elastic strip with edges rigidly clamped or free from stresses and which has N arbitrarily oriented defects of crack or rigid-inclusion type. A system of singular integral equations is constructed. The kernels in the equations is constructed. The kernels in the equations are expanded as series in a small parameter to derive a solution for a central crack or inclusion in a clamped strip. Graphs have been formulated relating the stress intensity coefficients around the defect vertices to the angle of orientation.The results can be used in calculating strength characteristics for composites reinforced by rigid fibers or coated with thin hard materials.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 26, No. 4, pp. 65–75, July–August, 1990.     

Temperature Fields in Oil-Gas Beds Exposed to the Action of an Acid

An asymptotic solution of the problem of a temperature field in carbonate-containing oil-gas beds exposed to the action of an acid is obtained. Expressions for the concentration fields of the acid and reaction products and the porosity and temperature in the zero and first approximation have been constructed. The space-time distributions of the temperature have been calculated, and the contribution of various physical processes has been analyzed.__________Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 2, pp. 51–64, March–April, 2005.