Mathematical models of fluid flow through non-deformable or elastically deformable porous media have become widely used. Especially they are common in water resources and oil drilling problems. The proposed hydrodynamic model of fluid filtration through plastically deforming porous skeleton, finds application in the investigation of the rapeseed extrusion extraction process. During this process the porous skeleton undergoes large inelastic deformation, which requires an alternative approach to the material behavior description. The material was presented by a two-component mixture of plastically deformable compressible porous medium saturated by oil. The mixture components were assumed to be nonreactive. According to the polymers extrusion processing theory, the problem was considered in reverse motion, the screw channel was unrolled on the plane, and viscous fluid model was used as governing equation for both mixture components. Further problem formulation was performed in the framework of the Euler motion description approach in a two-dimensional formulation for the plane of screw channel middlesection. Boundary-value problem formulated based on momentum balance and mass conservation equations for each mixture component. The boundary value problem independent variables are the mixture pressure, oil pressure, mixture velocity and oil velocity. The hypothesis of extraction speed proportionality to the oil pressure allows obtaining an approximate analytical solution for a constant filtration and compressibility coefficients.

The paper is devoted to theoretical investigation of the influence of stress state parameters of healed fault zones fragments on the characteristics of their mechanical response under shear deformation in the conditions of nonequiaxial compression. Investigation was based on computer-aided simulation by the movable cellular automaton method. Dimensionless parameter, named the degree of nonequiaxiality of compression was used as the basic parameter of stress state of the medium. This parameter characterizes the ratio of lateral and normal stresses in the plane of deformation. The main objective of the paper was to analyze the dependences of the shear strength, ultimate shear strain and value of volume changing (dilatancy) on the degree of compression nonequiaxiality at the initial stage of activization of the fault zone fragment. It is shown that the degree of compression nonequiaxiality of the medium is an important factor affecting the conditions under which healed fault zone could be activated. Here, the value of the shear stresses acting in a fragment of the medium, as well as corresponding ultimate values of shear strain and dilatancy, at which the fault zone could be activated, are essentially dependent on the ratio and dynamics of change of the local values of some stress tensor invariants. Among them are pressure and stress intensity (von Mises stress). This is due to the fact that these parameters determine the ability to operate in the geological medium of one of a key deformation mechanism that is associated with the formation and evolution of damages at the interface of the structural elements in the block structure medium. In particular, the decrease in the pressure in the medium fragment at relatively low levels of stress intensity can lead to an increase in ultimate shear strain and dilatancy at the beginning stages of activization of the fault zone. At the same time a significant increase in the stress intensity while decreasing the pressure could lead to a decrease in the shear strength of the geomedium.

In order to construct a theory that adequately describes the effects of cyclic loadings, it is initially necessary to analyze the experimental plastic loop of a hysteresis stainless steel SS304; and three types of backstresses responsible for the displacement of the center of the surface of loading are specified on this steel. For each type of backstresses we have formulated evolutionary equations on basis of the equations of the theory of plastic flow in the combined hardening. We have allocated the material functions which close the theory,. We have also formulated the basic experiment and method of the material functions identification. Evaluating the work of different types of backstresses on the field of plastic deformations under cyclic loadings with various magnitude of the deformation up to the experimental values of the number of cycles before failure, it has been obtained that the work of backstresses second type is a universal characteristic of the material. This result made it possible to formulate the kinetic equation of damage accumulation, based on which we have considered the nonlinear processes of damage accumulation. To determine the material functions responsible for the destruction, we have formulated the basic experiment and identification method. The authors have given material functions for stainless steel SS304. We have investigated the processes of elastic-plastic deformation of stainless steel SS304 with non-stationary hard cyclic loading under block changes of amplitude and mean deformation of the cycle. Also the processes of soft non-stationary and non-symmetric cyclic loading (ratcheting) under block changes of amplitude and mean stress cycle have been examined. The results of calculations are compared with the experimental results. Computational research of nonlinear processes of damage accumulation and low cycle fatigue of stainless steel SS304 are conducted under symmetric hard cyclic loading both at the constant amplitude of strain and block change of the amplitude of strain. The calculation results show that the scope of deformation reduction leads to increase of the nonlinearity of damage accumulation, while the increase of the deformation scale results in the fact that the accumulation of damages tends to be linear. There is a significant deviation from the rule of linear summation of damages under a satisfactory conformity of calculation results with the experiments. The paper presents such new results as: - specifying three types of backstresses responsible for kinematic hardening analyzing the experimental loops of plastic hysteresis; - establishing the work universality of the second type backstresses under low-cycle and high-cycle fatigue on basis of experimental results analysis; - constructing the theory of plastic flow under combined hardening and kinetic equations of damage accumulation on the basis of the evolution equations for three types of backstresses; - identifying the material parameters and verifying the proposed theory.

At this paper the algorithm of the subprogram for solving a two-point boundary value problem for system of the nonlinear differential equations of the first order is presented. The new algorithm of the subprogram named KLPALG united in itself the main ideas of the subprograms BVPFD (DD14AD, PASVA3) and PASSIN realizing the technique of continuation of solution by parameter. Besides, the generalized results of works of authors in a problem of nonlinear dynamic deformation of a thin spatial curvilinear rod calculated by its differential model are presented. The unknown functions in the equations of motion are calculated at discrete mesh points. The methods of direct integration allow us to express time derivatives by the current coordinates and coordinates and velocities calculated in the previous time steps. The first derivative of coordinate is replaced by finite difference; boundary conditions are added. The obtained system of nonlinear algebraic equations is solved by Newton method with the step length control of the convergence conditions. The Jacobi matrix of this system is of the block-tridiagonal structure which lends itself to efficient LU-decomposition. This decoupling of the Jacobi matrix allows you to quickly solve the corresponding system of linear algebraic equations of the big sizes. If the condition of convergence of Newton's method gives too small step, then used the technique of continuation of the solution on the parameter a (pseudo arc-length). As soon as the system of nonlinear equations is solved, to refine the nodal values of the calculated functions we use the deferred correction method. This method subtracts from the received solution the mistakes made by the approximation derived by the method of finite differences in the initial phase of the numerical solution. Thus obtained numerical solution is of accuracy appointed by user. This method is implemented in KLPALG subroutine which algorithm is presented in this paper.

Hooke’s law (in a modern tensor form considering different types of material anisotropy, finite or velocity formulation) is widely used in solid mechanics including physical and/or geometrical nonlinear problems. In the recent decades it has also been used in the majority of multilevel models oriented on describing inelastic deformation in mono- and polycrystalline materials. As a rule, in this case Hooke’s law is written using symmetrical measures of stress and strain state that are determined in terms of actual, intermediate (unloaded) or reference configuration. For a material that is elastic according to Green, the elastic potential presence naturally leads to the symmetry of elastic four-valent tensor П in the first and second pare of indices, П ijkl = П klij . However tensor symmetry in the first and second pair of indices is explained only due to the accepted and established agreement in solid mechanics related to symmetry of stresses and strains tensors. It is worth mentioning that the initial Hooke’s law written for uniaxial loading obviously had nothing to do with the symmetry of properties. The specified agreement made it possible to reduce the number of experiments necessary to find tensor elastic properties; and it is especially important for materials studies with an a priory low or unknown symmetry. Stress tensor symmetry results from law of conservation of angular momentum without distributed volume and surface moments. The neglection of the distributed surface moments is based on a hypothesis that two parts of the body interact with distributed forces, which can be put in to the stresses vector on each surface element. This hypothesis again is based on an idea that there is no correlation of distributed surface loadings on any material area element. It is worth stating that already in 1887 V. Voigt suggested to abandon this idea and put the distributed effects of one body part on the other one on any surface element into stresses vector and distributed moments vector. The specified suggestion is in a full compliance with the method related to putting a random system of forces into the principal vector and principal moment (this method is used in theoretical (classical) mechanics). The problem of a simple shear shows that Hooke’s (symmetrical) law leads to the incompliance of the stress state (found with the law in its conventional formulation) and part of boundary conditions. We have considered Hooke’s law which is oriented on application of asymmetrical measures of stresses and strains and elastic properties tensor with symmetry only in a pair of indices. Asymmetrical Cauchy tensor is used as a stress measure, gradient of displacement velocity (displacement velocities with respect to a stiff moving coordinates which is in charge for a rigid displacement of volume element) - as strain velocity measure; all of them do not depend on the reference coordinate. A type of tensor of elastic properties in Hooke’s law oriented on asymmetric measures is proposed.