Nonlinear problem of unsaturated frozen soil thawing in the presence of capillary forces

The problem of ice melting in unsaturated frozen soil in the presence of the capillary pressure in the water-air zone is formulated. The complete system of boundary conditions on the phase transition front is derived. For solving the nonlinear problem a numerical method is proposed. The dependence of the water saturation distribution on the form of the Leverett function, the capillary pressure, and the external pressure and temperature gradients is investigated. In the limiting case of saturated frozen soil the numerical and analytical solutions are compared.     

Note on the Radial Deformation and Stresses in Anisotropic Nonhomogeneous Elastic Media

Abstract

In this paper we study the elasticity problem of a cylindrically anisotropic, elastic medium bounded by two axisymmetric cylindrical surfaces subjected to normal piessures (plane strain). The material of the structure is orthotropic with cylindrical anisotropy and, in addition, is continuously inhomogeneous with mechanical properties varying along the radius. General solutions in terms of Whittaker functions are presented. The results obtained by St. Venant for a homogeneous cylindrically anisotropic medium can be deduced from the general solutions. The problem of a solid cylinder of the same medium under the external pressure is also solved as a particular case of the above problem. Problems of the type covered in this paper are encountered in nuclear reactor design.     

On large axisymmetrical deflection states of spherical shells

A boundary layer analysis is carried out to determine the possible form of large axisymmetrical deflection states for thin elastic spherical shells subjected to uniform external pressure. For the case of complete spheres it is shown that the governing equations admit boundary layer solutions corresponding to large deflections provided the pressure is sufficiently small. However, such solutions are found to exist for nonshallow clamped spherical caps for a much wider range of pressure. Numerical results are presented for the latter case.     

Buckling of long thick-walled circular cylindrical shells of isotropic incompressible hyperelastic materials under uniform external pressure

For isotropic incompressible hyperelastic materials, the problem of determining the critical external pressure at which a long thick-walled circular cylindrical shell will buckle involves solving a fourth-order system of highly non-homogeneous, ordinary differential equations. Closed-form solutions of this system are derived here for plane-strain conditions and for the particular case of the Varga material. These solutions are used to derive the buckling criterion and numerical values are obtained for the resulting critical pressures. They are found to be in good agreement with existing theoretical and experimental results for the neo-Hookean material.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Self-similar solutions of the problem of displacement of one gas by another in an axisymmetric case with a quadratic drag law

A problem of piston-induced displacement of one gas by another in cracks (porous media) in an axisymmetric case with a quadratic drag law is studied. Self-similar solutions for determining the dynamic characteristics (velocity and pressure) of the displacing and displaced gases are constructed in quadratures. The velocity and pressure are studied as functions of a self-similar variable for several initial conditions and parameters. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 87–92, September–October, 2008.     

Upper bound limit analysis of active earth pressure with different fracture surface and nonlinear yield criterion

Conventional calculations of static and seismic active earth pressures of soils on a retaining wall are formulated assuming the soils obeying a linear Mohr–Coulomb yield criterion. However, experimental evidences show that the strength envelopes of almost all geomaterials are nonlinear in nature over a wide range of normal stresses. In this paper, the strength envelope of the backfill behind a retaining wall is considered to follow a nonlinear yield criterion. A simple method is proposed for calculating the static and seismic active earth pressures acting against a retaining wall using a nonlinear yield criterion. This method is based on the upper bound theorem of limit analysis. Both translational and rotational fracture surfaces are employed in the formulation for calculating active earth pressures. Quasi-static representation of earthquake effects using a seismic coefficient concept is adopted for seismic active earth pressure calculations. Instead of using directly the actual nonlinear yield criterion, a linear Mohr–Coulomb yield criterion, which is tangential to the nonlinear yield criterion, is used to formulate the active earth pressure problem as a classical nonlinear programming problem. A nonlinear sequential quadratic programming algorithm is used to search for the maximum solution. In order to assess the validity of the proposed method, values of active earth pressures for different values of seismic coefficients and nonlinear parameters in the yield criterion are calculated and compared with solutions obtained using an extended Rankine’s active earth pressure theory. For the case of static active earth pressure, the upper bound solutions using the present method with a translational fracture surface are equal to the extended Rankine’s theoretical solutions and are slightly smaller than those obtained using the present method with a rotational fracture surface. For the case of seismic active earth pressure, numerical results obtained using the present method with a rotational fracture surface is very close to the extended Rankine’s theoretical solutions. A study is conducted to investigate the effects of the parameters in the nonlinear yield criterion on the active earth pressures.     

Optimal control and analysis of stabilometric tests with step disturbances of human sensory systems

Several optimal control problems are considered for a human motion model in the case of stabilometric tests when the posture of a person changes under a step disturbance. Such a situation is observed in the case of a galvanic vestibular stimulation or a vibration stimulation of the proprioceptors of shin muscles. Stabilometric tests with visual step disturbance are widespread in Russia. A characteristic feature of center of pressure (COP) trajectories for these tests is the presence of swings and overshoots. The test results are compared with the solutions of optimal control problems devoted to the repositioning of an inverted pendulum by a coupling torque. A speed-of-response problem and an optimal stabilization problem with a quadratic quality criterion are considered. The solutions to these optimum problems are compared with the results of stabilometric tests, which allows one to state a hypothesis on the sources of the above effects.     

Pressure distribution in the neighborhood of a growing fracture with a constant wedge force

Exact solutions of the problem of the pressure field in the neighborhood of a hydraulic fracture developing in accordance with a square root law in a permeable porous medium with a constant wedge force acting on the fracture edges are constructed. A particular case admitting a self-similar formulation and an exact solution and, as a result, the fairly complete investigation, is considered. The solution constructed holds for an arbitrary self-similar pressure distribution over the fracture edges. The problem considered reduces to the solution of a mixed boundary-value problem for the Helmholtz equation. The solution found can be useful both in itself and for testing more universal numerical algorithms.     

Uniqueness and stability of rigid-plastic spherical shells subjected to finite deformation under hydrostatic pressure

The rate problem for rigid-plastic strain-hardening deformations in structures subjected to prescribed hydrostatic pressure surface load is stated rigorously with due account of finite deformations. From the basic theory, a complete solution for admissible stress and velocity fields occurring at bifurcation is obtained for the problem of a spherical shell under arbitrary combinations of internal and external pressures. An earlier proven, sufficient condition for the uniqueness of continuing quasi-static deformation of a spherical shell is shown to be one of necessity. In the case of solely external pressure, it is shown that buckling modes are excluded by attention to an isotropically strain-hardening material with a non-singular yield surface. For preponderant internal pressure, however, it is possible for the predicted bifurcation mode to occur under increasing pressure.     

Determining the shape of an axisymmetric body in a viscous incompressible flow on the basis of the pressure distribution on the body surface

A method is developed for determining the shape of an axisymmetric body on the basis of the pressure coefficient distribution specified along the meridional section of the body. Viscosity is taken into account within the framework of the boundary layer model. The method is based on an iterative process, which involves the solutions of the inverse problem in the plane case and of the direct problem for an axisymmetric body. A code implementing the iterative process is written, and examples of numerical results are given.     

Forced oscillations of a gas bubble in a spherical volume of a compressible liquid

A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed. Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 111–118, March–April, 1999.     

Exact solutions of the kinetic-moment equations of a mixture of monatomic gases

An extension is given of the class of exact solutions of the kinetic-moment equations for a monatomic gas in the absence of external forces [1] to the case of a mixture of monatomic Maxwellian gases with account for external forces. Very simple solutions of this class are obtained which are examples of the normal solutions of the Boltzmann equations in the Chapman-Enskog sense [2]. Conclusions are summarized concerning the applicability of the various methods of solving the Boltzmann equations and their properties, obtained on the basis of an analysis of the indicated exact solutions.The author wishes to thank M. N. Kogan and A. A. Nikol'skii for their interest in the study.     

Heterogeneous flows of multi-component mixtures in porous media—review

The main difficulty in the mathematical simulation of two-phase flows of gas-condensate or gassed oil in reservoirs is connected with a choice of binary or ternary models of hypothetical components representing the real multi-component mixture. The reduction of the number of degrees of freedom is illustrated by a plot of dependence of Gibbs concentration parameter on pressure for some numerical solutions. These solutions are calculated on the basis of balance equations, generalized Darcy law and equilibrium phase composition for a system of methane-n-butane-decane. It is shown that a binary model is adequate for the simulation of steady, quasi-stationary and self-similar flows into a well. Mathematical simulations of well-capacity for steady flows and processes of pressure build-up are considered.The mathematical problem of gas-condensate driven by dry gas is formulated. The corresponding solution with discontinuities is discussed by means of balance laws for jumps. It is shown that in the case of gas cycling processes a ternary model is necessary as the simplest one.The different approaches to the problem are also discussed.     

Weak Solutions for a Class of Nonlinear Systems of Viscoelasticity

The principal focus of the article is the construction of classical weak solutions of the initial value problem for a class of systems of viscoelasticity in arbitrary spatial dimension. The class of systems studied is large enough to incorporate certain requirements dictated by frame indifference and also has a structure which allows for a variational treatment of the time-discretized problem. Weak solutions for this system are constructed under certain monotonicity hypotheses and are shown to satisfy various a priori estimates, in particular giving improved regularity for the time derivative. Also measure-valued solutions are obtained under a uniform dissipation condition, which is much weaker than monotonicity. A special case of the viscoelastic system is the gradient flow of a non-convex potential, for which measure-valued solutions are here obtained, a new result in the vectorial case. Furthermore, in this setting it is possible to show that these measure-valued solutions satisfy a certain property which ensures they coincide with the classical weak solution when this exists, as for example in the convex case where existence and uniqueness are well known. Accepted July 1, 2000?Published online December 6, 2000     

Positive Solutions to Singular Second and Third Order Differential Equations for Quantum Fluids

We analyze a quantum trajectory model given by a steady-state hydrodynamic system for quantum fluids with positive constant temperature in bounded domains for arbitrary large data. The momentum equation can be written as a dispersive third-order equation for the particle density where viscous effects are incorporated. The phenomena that admit positivity of the solutions are studied. The cases, one space dimensional dispersive or non-dispersive, viscous or non-viscous, are thoroughly analyzed with respect to positivity and existence or non-existence of solutions, all depending on the constitutive relation for the pressure law. We distinguish between isothermal (linear) and isentropic (power law) pressure functions of the density. It is proved that in the dispersive, non-viscous model, a classical positive solution only exists for “small” (positive) particle current densities, both for the isentropic and isothermal case. Uniqueness is also shown in the isentropic subsonic case, when the pressure law is strictly convex. However, we prove that no weak isentropic solution can exist for “large” current densities. The dispersive, viscous problem admits a classical positive solution for all current densities, both for the isentropic and isothermal case, with an “ultra-diffusion” condition. The proofs are based on a reformulation of the equations as a singular elliptic second-order problem and on a variant of the Stampacchia truncation technique. Some of the results are extended to general third-order equations in any space dimension. Accepted July 1, 2000?Published online February 14, 2001     

Axisymmetric deformations of circular rings made of linear and Neo-Hookean materials under internal and external pressure: A benchmark for finite element codes

The axisymmetric deformations of thick circular rings are investigated. Four materials are explored: linear material, incompressible Neo-Hookean material and Ogden's and Bower's forms of compressible Neo-Hookean material. Radial distributed forces and a displacement-dependent pressure are the external loads. This problem is relatively simple and allows analytical, or semi-analytical, solution; therefore it has been chosen as a benchmark to test commercial finite element software for various material laws at large strains. The solutions obtained with commercial finite element software are almost identical to the present semi-analytical ones, except for the linear material, for which commercial finite element programs give incorrect results.     

Numerical calculation of the motion of a gas from a surface explosion

The motion of a gas by the normal impact of a high-speed body at the interface between a dense half-space and a vacuum is investigated numerically. The motion of the shock wave and the shape and distribution of the parameters of the gas dispersing in the vacuum are obtained. The motion is studied during the formation of a region with high pressure at the boundary with the vacuum of a gas occupying the half-space z > 0. The assumption of cylindrical symmetry relative to the z axis enables this three-dimensional nonsteady-state problem in the general case to be solved as a two-dimensional problem. For the corresponding one-dimensional problem, the numerical solution and, for certain gases also, the analytic solutions are well known and are considered in detail in [1]. As a result of solving the two-dimensional problem, profiles of the gasdynamic quantities are obtained which are similar to the solutions in the one-dimensional case and the result of the solution by a self-similar method. The cup-shaped surface of the shock wave front with a pressure gradient on it “focusses” the dispersing gas so that its velocity component normal to the surface z = 0 is greater by an order of magnitude than the component parallel to the surface of separation of the medium, and only at individual points is their ratio close to 0.4. Therefore, the dispersing gas is formed into the shape of a “jet”, the pressure and density profiles on the axis of which have a shape similar to the one-dimensional problem of a brief shock, but in the plane z = 0 the pressure and density distributions are similar to the distributions of these quantities in the case of a powerful point explosion in an unbounded medium. The initial disturbance in the symmetrical problem being considered may be the result of either the normal impact of the body with a high velocity at the surface of the dense medium, or the consequence of the effect of a giant laser pulse, or some other process when a certain volume is formed with a high pressure at the interface between the dense medium and a vacuum, or with another low-density medium.     

Normal flow boundary conditions in 3D circulation models

The problem of enforcing normal transport conditions on 3D velocity fields is considered in the context of ‘wave equation’ finite element models. A procedure for strong enforcement of the transport constraint is given. The procedure is identical for Neumann (transport known) and Dirichlet (pressure known) problems, which are treated reversibly. All local mass and force balance relations are retained in the FEM system. A global mass conservation property is proven for the general 3D, discrete-time case. Examples demonstrate the quality of the solutions and the practicality of the approach. © 1997 John Wiley & Sons, Ltd.     

Two-dimensional elastica analysis of equilibrium shapes of single-anchor inflatable dams

Several types of inflatable dams are considered. These are long, air-inflated, cylindrical structures on a rigid foundation. Sometimes one of the long edges of a sheet is folded back to the other edge, and then the two edges are clamped to the foundation along a single anchoring line. A second configuration can be modeled as two sheets attached along two long edges, with one edge anchored and the other free to lift as air pressure is applied between the sheets. Another device treated here is a hinged spillway gate lifted by an inflatable bladder. The cross section of the dam or bladder is analyzed as an inextensible elastica. The governing equations and boundary conditions are formulated for each case, and shooting methods are utilized to obtain numerical solutions for the equilibrium shapes. The effects of the internal air pressure and the external water height are investigated.