Mixed forced, parametric, and self-oscillations are considered if there is a delay in the elastic force in the system. A dynamic model is a friction self-oscillation system describing the frictional self-oscillations that occur in many technical systems for various purposes (metal-cutting machines, textile equipment, brakes and a number of other engineering objects). The operation of the system is supported by the energy source of limited power. For the analysis we used the method of straight linearization which is easier than the known methods of analysis of nonlinear systems, has no time-consuming and complex approximations of different orders, provides an opportunity to obtain the final design ratios regardless of the specific type and degree of nonlinearity, thus reducing labor costs and time by several orders of magnitude. By using this method, we obtained solutions of a nonlinear system of differential equations describing the system's motion. The equations of non-stationary and stationary movements are derived. To analyze the stability of stationary movements, the stability conditions based on the Routh-Hurwitz criteria are compiled. Calculations were performed to obtain information about the effect of delay on the oscillation modes. It is shown that the delay affects both the magnitude of the amplitude and the location of the amplitude-frequency curve in the frequency range depending on the magnitude of the delay, the amplitude curve is shifted to the region of lower frequencies. The stability of stationary oscillations depends both on the energy source characteristics and lag value. The interaction of the oscillating system and the energy source leads to a number of effects, both in the presence and absence of the lag. However, their course may be different depending on the lag value.

Postcritical deformation of a material is a process which is characterized by a decrease of stress during growing deformations as a result of accumulation of structural damage. The design becomes unable to withstand the external load only when zones with weakened connections are developed enough. Evolution of postcritical deformation zones can occur with an increase of the external load applied to the construction. It means that taking into account the softening of the material allows determining the strength and deformation reserves of constructions more accurately. The mathematical formulation of the boundary value problem of supercritical deformation mechanics is given in the paper. The features of the experimental study of the postcritical stage of material deformation are listed. Strain curves of various steels with a long section of softening are obtained. Numerical solutions for the problems of deforming a thin plate with stress concentrators of different geometries under kinematic loading are obtained. Piecewise linear approximations and real strain curves of steel 20 and steel 40Cr4 obtained experimentally are considered. The evolution of zones of postcritical deformation in the material is considered. The correspondence between the value of the decline modulus and the nature of the evolution of the softening zones is determined. A stress plot is constructed that reflects how the complete material deformation diagram is realized near the concentrator. The calculated loading diagrams are constructed. It is noted that even after the appearance of softening zones, an increase in external load is possible. The strength and deformation resources of structures are determined, and the influence of the geometry of the stress concentrator on their values is considered. It is noted that the consideration of softening in modeling the behavior of structures with stress concentrators is appropriate.

This paper defines the structural strength criterion for 4DL-reinforced carbon-carbon materials. For this scheme, fiber reinforcement consists of four groups of reinforcing elements, three of them are located in parallel planes with the angles of 120° between them and the fourth one is normal to them. The paper addresses the first failure of the material corresponding to its yield stress, in this point, one of the material components deviates from linear elastic behavior. A composite material is considered to be non-uniform structurally and consists of a matrix and reinforcing elements, rods. Those rods, in their turn, represent a unidirectional composite. To analyze the stress-strain state of individual components of the material, a three-level elastic model is built that uses the analytic approach at the micro level, while at higher levels it uses the finite element method. For numerical calculations, a structural cell of the material is taken. The boundary conditions provide small to negligible influence of the edge effects, thus simulating the behavior of the infinite volume of the material. For the material components, local strength criteria are introduced, where the fields of the criterion quantities are averaged over the volume of the structural cell. The strength surface of the material that corresponds to its first failure is obtained, and the conclusion is made that the suggested criterion provides a reasonable agreement with the available data on the typical carbon-carbon composite characteristics. Based on the calculated dependencies of the material’s yield stress on the load direction, a procedure is suggested to identify the model parameters based on the material failure behavior analysis using standard tensile and compressive tests. Estimated discrepancies between the results calculated using the suggested criterion and those obtained using the limiting stress criterion for biaxial stress states are given. It is shown that the discrepancy may reach tens of percent and in some cases the material strength increases in comparison with that in the uniaxial stress state. The results are subject to verification tests in order to verify the model for advanced spatially reinforced carbon-carbon composite materials.

In the paper presents the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for a full range of the mode mixities. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities both for plane strain and plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions, all values of the mixity parameter within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained using the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher 0.89 respectively for plane strain conditions and seven sectors determine the asymptotic solution for the mixity parameter less than 0.39, while five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain. For plane stress conditions this segment can’t be observed. The solution in each sector corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials.

This paper presents a calculation and experimental technique for determining stress intensity factors in an imitation model of a titanium alloy disk. We studied a low-pressure compressor disk of a gas turbine engine (GTE) D-36. During operations, there occur fatigue cracks initiated and developed in the slot fillet under the blade at the place of transition of the bottom to the lateral surface of the inter-groove projection, which lead to a separation of the disk’s part within its rim. The mixed-mode crack growth occured in the compressor disks. Based on the principles of imitation modeling, the geometry and loading condition of the imitation model of the compressor disk was developed. The fatigue test of the imitation model was carried out with a frequency of 5 Hz, at room temperature and with stress ratio Rc = 0.1, by means of a biaxial testing machine. The crack growth was monitored using an optical microscope. The criterion for failure was the condition for reaching a growing crack of the compensation hole. During the test, the positions and sizes of the crack fronts were fixed, which are the basis for the numerical calculation of the fracture resistance parameters. In the order of the numerical studies, six three-dimensional finite element models with different positions and sizes of the crack fronts are considered. The results of the numerical calculations based on the finite element method were used to determine the distributions of elastic and plastic stress intensity factors along each crack front. We demonstrated the advantages of the calculation and experimental methods for solving the problems of interpretation and prediction of the crack growth in the rotating disks of turbomachines using the methods of fracture mechanics.

Verification of the structures operating possibility using numerical modeling beyond the elastic limit requires standardization of safety factors and calculation methods used to get them. In the framework of the discussion on the improvement of the strength standards of the aviation and nuclear industries for structures operating under low-cycle mechanical and reversible dilatation (temperature, hydrogen) external influences, the article discusses the limiting states; the deformation properties of materials necessary for their calculation; safety factors for loads and durability; calculation methods. The article divides limit states of structures under low-cycle actions into two groups: typical, corresponding to a qualitative change in the deformation type, and individual, determined by allowable displacements and cracks for a particular structure. The following types of deformation are considered: inelastic deformation only at the running-in stage (that changes to elastic after the auspicious residual stresses develop and cyclic hardening of the material); alternating flow (that continues with the number of cycles); progressive accumulation of strains and displacements; combined deformation (when both strain span and strain increment are non-zero in a stable cycle). The types of deformation differ in possible consequences for the structure and the initial data for the calculation: mechanical properties of the material required for modeling different types of deformation should be determined by fundamentally different tests. An analysis of individual limit states without taking into account differences in the types of deformation - and thus typical limit states - may be incorrect. The main focus of the article is on typical limit states. The limit states vary depending on the stage of operation at which inelastic cyclic deformation is allowed. Inelastic deformation expands allowable load range, the expansion due to the inelastic deformation at the running-in stage only is usually more significant than additional expansion due to the continuous inelastic deformation; besides, the inelastic deformation only at the running-in stage does not demand analysis of low-cycle fatigue and accumulated strains. Further expansion of the permissible load range, as well as solution of safety problems based on risk assessments, requires a more complete study of the deformation properties of materials at the pre-fracture stage, where cyclic softening predominates.