Complex Stress State Research Articles

TRIAXIAL STRESS STATE OF CONCRETE UNDER LONGITUDINAL DEFORMATION OF TUBE-CONCRETE SAMPLES

The effects that occur during the transition of concrete to a complex stress state, resulting from the limitation of transverse deformations due to the steel shell, are considered using experimental modeling. The studies were carried out on CFST samples of three standard sizes, as well as on differentiated samples of pure concrete and steel tube. Diagrams of work of samples under longitudinal compression and their comparison are given. The use of CFST structures turns out to be very effective not only as a result of saturation of the supporting structure with material, but due to the mutual influence of the components of the structure on each other. A quantitative assessment and a qualitative justification for increasing the efficiency of the concrete core as part of a CFST element are given. It is shown that the increase in the bearing capacity of the sample due to the reinforcement of a simple tube with concrete is several times higher than the differentiated bearing capacity of the concrete core. This fact makes it possible to consider CFST structures as composite ones with a non-linear increase in strength due to the joint work of the system components. The transition of specimens to the plastic state of tube-concrete specimens occurs at the same values of longitudinal deformation as for hollow steel tubes, but at high load values. The effects that occur in concrete under the influence of a steel shell create a triaxial stress state, which makes it possible to multiply the bearing capacity of any brittle material. It is shown that concrete under conditions of such a stressed state behaves rather like an amorphous material, does not collapse at all stages of deformation, and there are practically no processes of crack initiation and its propagation in concrete.

Sound synthesis approach based on the elastic stress analysis of a wrinkled thin film coating

This paper examines the ability to employ the model of a wrinkled thin film coating under plane strain conditions for the generation and manipulation of sounds. It is assumed that the film-on-substrate structure represents a multilevel system consisting of four phases with different elastic properties such as surface layer, film material, interphase and substrate material. The undulation of surface and interface geometry leads to the complex stress state of the film coating resulting from the superposition of two perturbed stress fields evolved near the curved surface and interface. The analysis of the stress state reveals the periodic distribution of the longitudinal stresses smoothly changing from surface to interface. This facilitates the creation of a sound synthesis technique similar to the timbre morphing method which provides the transition between two waveforms while creating new intermediate waveform during this process. The perspective of using the wrinkled thin film model for sound generation is gained by its complex behavior, where the influence of one parameter on the stress distribution is affected by other parameters, which in turn reflects in the rich sound morphologies during the mapping of the stress oscillations onto sound.

FINITE ELEMENT ANALYSIS ON THE NONLINEAR BEHAVIOR OF THE RC SHEAR WALL WITH REGULAR OPENINGS INFLUENCED BY HIGH-STRENGTH STEEL

This paper presented a nonlinear finite element analysis of lateral loading RC shear walls with regular openings using the 3D-NLFEA program. The RC shear walls model was generated from the available test results in the literature. To model the concrete under a complex stress state, a multi-surface plasticity model which combines compression failure surface with tension cut-off failure surface was used. The model was intended to look at the load-displacement relationship and the crack pattern between the model and the numerical model. In addition to the numerical model verification, parametric studies were carried out to investigate the use of high-strength steel (HSS) of the two different grades (grades 100 and 120) to replace all the normal-strength steel (NSS) or only some of it. The parametric studies found that the shear wall with the NSS bar demonstrated higher stiffness and achieved higher lateral load with the lowest extent of damage (compared to the RC shear wall with the HSS bar). On the other hand, using the HSS bar resulted in lower stiffness, lower lateral load, and higher damage region, which was expected as more strain is required to yield the HSS bar.

Deformation mechanisms and microstructural characteristics of AZ61 magnesium alloys processed by a continuous expansion extrusion approach

The unique continuous extrusion-based severe plastic deformation approaches were proposed recently to process high-performance magnesium (Mg) alloys, while the in-depth deformation mechanisms under such complicated thermomechanical conditions were not well understood. In the present work, the fundamental deformation behaviors of AZ61 Mg alloy from 25 to 400 °C were firstly examined under uniaxial compression deformation. Then the deformation mechanisms and microstructural characteristics of AZ61 Mg alloy during continuous expansion extrusion forming (CEEF) were systematically investigated by microstructural observations, finite element and cellular automata simulations. The results showed that the continuous evolutions of temperature, larger strain level and complex stress state with strain rate range of 0 ∼ 5.98 s−1 during CEEF brought the distinctive dynamic recrystallization behaviors and texture development in AZ61 Mg alloy, which were different to that of uniaxial compression deformation. In details, a remarkable grain refinement was achieved via CEEF processing due to the simultaneous actions of continuous dynamic recrystallization (CDRX) and discontinuous dynamic recrystallization (DDRX). Gradually enhanced CDRX were observed from center to edge region, which had significant effects on the texture distribution and texture strength. The c-axis of most grains rotated under distinctive shear strain following parabolic metal flow, resulting in stable fiber texture. In addition, the evolution of the internal texture of the alloy led to an obvious increase in the Schmid factor for the activation of basal 〈c + a〉 slip system.

Influence of strain rate on grain refinement and texture evolution under complex shear stress conditions

The effects of different complex shear stress conditions on grain refinement and texture evolution of Mg-13Gd-4Y-2Zn-0.5Zr alloy were investigated. With increasing strain rate, the average grain size of compression-shear (CS) and compression-torsion (CT) samples are decreased, and the grain size of dynamic recrystallization (DRX) grains is also decreased. This is because that the precipitation number of β phases is increased, and the hindering effect on grain growth can be significantly enhanced. The DRX fractions of CS and CT samples are decreased with increased strain rate. The low DRX fraction at high strain rate is related to the insufficient time for grains to nucleate. The DRX process can be promoted by the PSN mechanism of second phases, and the grain growth can be restricted by the pinning effect. At the same time, the texture strength is enhanced as the strain rate increased. Besides, the kinking degree of lamellar long-period stacking ordered (LPSO) phases is increased. Under complex shear stress conditions, non-basal slip, especially pyramidal slip, is easily activated and the texture is deflected greatly. Compared with the CS samples, CT samples have smaller average grain size, higher DRX fraction, and lower texture strength for a certain strain rate. This is because that the equivalent stress of the CT sample is larger, the stress triaxiality is smaller, so more serious dislocations are piled up near grain boundaries and second phases. At the same time, since CT sample was sheared with torsion, the dislocation movement path can be called “rotational dislocation accumulation”, and the longer distribution path of the CT sample is generated, so more sub-grains and low-angle grain boundaries (LAGBs) are formed. Compared with the CS sample, more huge-angle grain boundaries (HAGBs) and DRX grains are formed from grain boundary to grain interior, so better grain refinement effect is achieved.

Constitutive relations, mechanical behaviour, and failure criterion of microcapsule-based self-healing concrete under uniaxial and triaxial compression

Microcapsule-based self-healing concrete can autonomously release repairing agents to bond and seal cracks when the material is damaged. It is seen as a promising construction material from the perspective of a sustainable society. In addition, it is important to study mechanical behaviour under complex stress conditions to promote the application of self-healing concrete in engineering structures. This study investigated the constitutive relations, strength and deformation, and failure criterion of the epoxy/urea-formaldehyde microcapsule-based self-healing concrete through uniaxial compression, conventional triaxial compression and true triaxial compression tests, in which the influence of different microcapsule dosages, confinement pressures and the principal stress ratios on the mechanical behaviours was considered. The results showed that the ultimate strength of the specimen decreased with increasing microcapsule dosage under all loading conditions. The lateral pressure significantly promoted the ultimate strength and deformation capacity of the specimen, and this increase mainly depended on the maximum principal stress (σ1) rather than the intermediate principal stress (σ2). By visual checking the failure modes of the specimens, more cracks were observed under the conventional triaxial compression than those under the true triaxial compression. This multi-crack failure mode allowed more core material of microcapsules to enter the microcracks, which accordingly increased the strength of the microcapsule-based concrete.

Analysis of force and deformation parameters in corrugated clad rolling

Corrugated clad rolling is an emerging and promising technology to produce composite plates with a three-dimensional corrugated interface and excellent performance. A new analytical model is proposed to predict and analyze force and deformation parameters in corrugated clad rolling using the slab method, which solves the problem that there is no suitable analytical model to improve the precision of automatic production control. The analytical model is proposed by selecting a differential element perpendicular to the rolling direction, analyzing the corresponding differential equilibrium equations and calculating the integral constants according to the yield and boundary conditions. Dynamic rolling forces under different contact states between the composite plate and rolls are presented creatively considering the special features of the new corrugated clad rolling that there are multiple complex shapes and stress states in the deformation zone and that there are various outlet conditions of the corrugated composite plate. The accuracy of the proposed analytical model is verified by the comparisons of the calculated values from the model, the FEM results, and the experimental measurements. The error of the proposed model is within 7.36%. The specific rolling pressure distributions along the projected contact arc, the specific horizontal stress distributions along the projected contact arc, the dynamic rolling forces, and the positions of the lower neutral point can be conveniently and accurately calculated by the proposed model, which can help to analyze the mechanism of the corrugated rolling with the variation of the rolling reduction rate, the shear yield stress ratio and the friction factor ratio. The proposed analytical model of dynamic rolling force and deformation parameters is especially suitable for the analysis of the new corrugated clad rolling and can provide technical support for the control precision of automatic production.

Modeling of both tensional-shear and compressive-shear fractures by a unified phase-field model

Many phase-field models have been developed in recent years to captue different fracture modes from tensional to shear, tensional-shear, and compressive-shear fractures. However, there seems no phase-field model that can simulate the tensional, shear, tensional-shear, and compressive-shear fractures at the same time under complex stress states. In this paper, a unified phase-field model is proposed in the framework of the original phase-field theory. A universal fracture criterion, that can predict both tensional-shear and compressive-shear fractures under complex stress states is embedded in the proposed phase-field, and the failure compression strength is introduced to consider the fracture under a compressive stress state. Therefore, the crack direction can be directly determined from the universal fracture criterion. The strain energy of undamaged configuration is decomposed into three parts, the tensional/compressive part, the shear part, and the rest part. The tensional/compressive and shear parts can be degraded by different degradation functions or the same degradation function. Cohesive fracture models with general softening laws and the classical brittle fracture model can be used in the proposed model, and the length scale has much less influence on the global response if cohesive fracture models with general softening laws are applied. Numerical examples show that the proposed model has the ability to simulate both the tensional-shear and compressive-shear fractures in rock-like materials and the results are in good agreement with the experiments.

Strengthening Modulus and Softening Strength of Nanoporous Gold in Multiaxial Tension: Insights from Molecular Dynamics.

The functionalized applications of nanoporous metals place clear requirements on their basic mechanical properties, yet there is a lack of research on the mechanical response under multiaxial loading conditions. In this work, the mechanical behaviors of nanoporous gold under multiaxial tension are investigated via molecular dynamics simulations. The mechanical properties under different loading conditions are compared and the microstructure evolution is analyzed to clarify the deformation mechanisms of nanoporous gold in biaxial and triaxial tension. It is found that the modulus of nanoporous gold in multiaxial tension is strengthened and the strength is softened compared to uniaxial tension. The failure of nanoporous gold in multiaxial tension is dominated by the progressive yielding, necking, and rupture of ligaments along the multiple uniaxial loading directions. The dislocation activity under multiaxial loads is more intense and more prone to plastic deformation, ultimately resulting in lower strength and smaller failure strain. The findings provide more insight into the understanding of the deformation mechanisms of nanoporous metals under complex stress states.

Discovery of multimechanisms of screw dislocation interaction in bcc iron from open-ended saddle point searches

Dislocation motion and interactions determine mechanical properties in body-centered cubic (bcc) metallic materials. However, studying mechanisms for the screw dislocation interaction is fundamentally challenging since many underlying processes involve mesotimescales and atomistic resolution, currently inaccessible by either experimental techniques or continuum theoretical methods. In this paper, we develop a computational capability based on self-evolving atomistic kinetic Monte Carlo (SEAKMC) to sample the critical events and saddle point energies related to screw dislocations and their junctions. The method is first validated by calculating the stress dependence of Peierls barriers and formation energies of kink pairs and cross-slip kink pairs on a single screw dislocation in bcc iron. Then the method is applied to a binary junction of a pair of intersecting screw dislocations, the structure of which is crucial for low-temperature plastic deformation. We identify three important mechanisms: coplanar cross-slipping, jog-pinning, and a previously unknown unzipping mechanism during the evolution of the binary junction. The mechanisms are then further validated using classical molecular dynamics simulations. The computational capability developed in this paper provides an effective tool to evaluate screw dislocation related thermally activated events in complex stress conditions. The mechanisms discovered in this paper provide critical insights into temperature dependence of the anomalous slip, a breakdown of the Schmidt law, during the plastic deformation in bcc iron and can be generalized to other bcc metals.

Isothermal Mechanical Cycling of Saponite–Titanium Composites in Conditions of Complex Stressed State

Abstract It has been experimentally shown that in the isothermal mechanocycling of continuous cylindrical samples of saponite–titanium composites under conditions of a complex stress state, a reversible deformation of the properties of martensitic inelasticity is observed, which appears during thermal cycling at intervals of martensitic transformations. According to the results of the experiment, the influence of the ratio between the static and cyclic components of the stress on the change of shear deformation in a complex stress state during mechanical cycling with axial load is estimated.

Theoretical Investigation of Forming Process of Aluminum Alloy Rail Vehicle Side Window

With the vigorous development of rail transit trains around the world and the emergence of global environmental pollution and energy shortages, the world has an urgent need for manufacturing technology for lightweight aluminum alloy rail transit train components. This paper mainly studied the superplastic forming law of 5083Al for rail transit. Through the high-temperature tensile test and blowing forming experiments, the superplastic properties of 5083Al were determined. Based on this, the die design, finite element simulation, and forming experiment of the rail vehicle side window were carried out. In order to study the superplastic deformation behavior of industrial 5083Al under complex stress conditions, the influence of the depth, area ratio, and friction coefficient of the pre-forming die on the part thickness distribution is simulated. The side window is made of a high-strength 5083Al sheet in the form of bending at both ends to ensure the strength of the connection between the overall side window and the side wall skeleton. The variation law of the side wall forming height of 5083Al box-shaped parts was studied. The efficient manufacture of parts that meet quality standards was made possible by the optimization of the pressure profile. The microstructure changes of the material after superplastic forming were studied by Energy Dispersive Spectrometer (EDS) and Electron Backscattered Diffraction (EBSD).

A ductile fracture model incorporating stress state effect

To understand comprehensively the relationship between material ductility and stress state, the new shear, shear-compression, and shear-tension specimens were specially designed, and seven types of specimens were used in total to achieve precise control of stress states over wide ranges of stress triaxialities and Lode angle parameters. A hybrid experimental-numerical method was employed to determine the equivalent plastic strain, stress triaxiality, and Lode angle parameter, and hence to construct the 3D fracture locus of the Ti-6Al-4V alloy. Different micromechanisms were found to dominate the failure process under various stress states. Based on the experimental results, a new ductile fracture model coupling both stress triaxiality and Lode angle parameter was proposed and implemented into the commercial finite element program ABAQUS/Explicit via the user material subroutine VUMAT. Comparative studies with some other fracture criteria indicate that the proposed fracture model can capture the non-monotonic feature of fracture strain, and predict the fracture strain with better accuracy in a wide range of stress states. A verification test was performed to check the predictive capability of the present model. The results show the excellent agreement between experiment and simulation with respect to the fracture displacement and fracture morphology. This study paves the way for characterizing and predicting the ductile behavior of metallic materials under complex stress states, and provides a fundamental databank for the analysis and design of Ti-6Al-4V alloy structures.

Mechanical behavior of bonded interface between steam-cured concrete and SCC under direct shear test: Experimental and numerical studies

The interlayer damage of slab ballastless track for high-speed railway has become a weak link that affects the safe operation of track structure. Under the combined action of the train self-weight load and wheel-rail friction, the track plate presents complex stress state. Meanwhile, the interlayer defects and bond state have a great influence on the damage mechanism of track plate. The failure mechanism of concrete-concrete interface under combined stress has been focused on, but it is still not well quantitatively investigated at the grain scale. In this paper, the discrete element method (DEM) combined with digital speckle and acoustic emission monitoring technology is used to conduct the direct shear tests on bonded samples composed of steam-cured concrete and self-compacting concrete (SCC) with different bond interface characteristics. On this basis, it is quantitatively analyzed the effects of interfacial bond strength and interfacial morphological characteristics on the compressive-shear mechanical properties and damage patterns of bonded samples, and the deterioration mechanism of the interface is explained from a mesoscopic perspective. The results show that the compression-shear mechanical response and damage features of the bonded samples are the result of the combined effect of bond strength and morphological characteristics of the interface. The increase of bond strength makes the samples change from shear cracking damage along the interface to tensile cracking damage inside the concrete. The bonded area ratio (BAR) of the interface maintains a good positive linear correlation with the shear strength, and the increase of BAR makes the brittle failure more obvious in the post-peak stage. The increase of interface roughness coefficient (IRC) makes more compressive stress concentration areas form in the interface, which enhances the shear strength of the bonded sample to a certain extent.

Experimental study of fatigue load effect on the hydraulic fracturing behavior of granite

AbstractRock masses around fluid injection projects are usually subject to complex stress states, including hydraulic pressure, in situ stresses, and fatigue loads. Thus, a series of hydraulic fracturing experiments were performed on granite to simulate such stress states using the true triaxial dynamic testing system. Computed tomographic (CT) imaging was performed to identify the fatigue effects on hydraulic fractures. The results indicate that the increasing amplitudes of cyclic load applied on the minimum principal stress direction will change fracture initiation pressure and generate nonplanar and narrow fractures. When the disturbance direction was changed to the intermediate principal stress, the higher amplitude corresponds to the lower breakdown pressure and the shorter pressurize duration and leads to wider fractures. With the increasing disturbance frequency, complex fatigue cracks were generated, which might weaken the rock strength. The present experiments can enhance the understanding of the hazard of fatigue loads on hydraulic fracturing.

Modeling Progressive Damage and Failure of Single-Lap Thin-Ply-Laminated Composite-Bolted Joint Using LaRC Failure Criterion.

Thin-ply composite failure modes also significantly differ from conventional ply composite failure modes, with the final failure mechanism switching from irregular progressive failure to direct fracture characterized by a uniform fracture with the reduction of the ply thickness. When open holes and bolt joints are involved, thin-ply-laminated composites exhibit more complex stress states, damage evolution, and failure modes. Compared to the experimental study of thin-ply-laminated composite-bolted joints, there are few reports about numerical analysis. In order to understand the damage evolution and failure mechanism of thin-ply-laminated composites jointed by single-lap bolt, a progressive damage model based on three-dimensional (3D) LaRC failure criterion combined with cohesive element is constructed. Through an energy-based damage evolution method, this model can capture some significant mechanical characteristics in thin-ply-laminated structures, such as the in situ effect, delamination inhibition, and fiber compressive kinking failure. The comparisons between the numerical predictions and experimental observations are made to verify the accuracy of the proposed model. It is found that the predicted stress-displacement curves, failure modes, damage morphologies, etc., are consistent with the experimental results, indicating that the presented progressive damage analysis method displays excellent accuracy. The predicted stress at the onset of delamination is 50% higher than that of the conventional thick materials, which is also consistent with experimental results. Moreover, the numerical model provides evidence that the microstructure of thin-ply-laminated composite performs better in uniformity, which is more conducive to inhibiting the intra-layer damage and the expansion of delamination damage between layers. This study on the damage inhibition mechanism of thin-ply provides a potential analytical tool for evaluating damage tolerance and bearing capabilities in thin-ply-laminated composite-bolted joints.

Investigating the failure behavior of hygrothermally aged adhesively bonded CFRP-aluminum alloy joints using modified Arcan fixture

Composite adhesively bonded structures used in automobiles usually withstand the harsh environment and complex stress during long service. To further popularize the carbon fiber reinforced polymer (CFRP), it is necessary to study the failure behavior of environmentally aged CFRP-aluminum alloy joints under complex stress state and establish a mechanical performance prediction method. In this work, quasi-static tensile tests on hygrothermally (80 °C and 95% relative humidity) aged adhesively bonded CFRP-aluminum alloy butt joints were carried out using a modified Arcan fixture. The stress distribution in the adhesive of Arcan tests was analyzed by the FEM, which proved the rationality of the designed fixture. After exposure to hygrothermal aging for a maximum of 30 days, the mechanical performance of the joints under different stress states decreased remarkably at a rapid degradation rate during the early stage. For unaged joints, mixed adhesive–cohesive​ failure and CFRP fiber tearing were observed when normal stress was predominant, whereas complete cohesive failure in the adhesive occurred when shear stress was more significant. For the hygrothermal aged joints, the failure modes under different stress states were adhesive–cohesive failure because of the significant degradation in adhesive performance. The q-stress failure criteria and environmental degradation factors of the CFRP-aluminum alloy bonding structure before and after aging were established, and both were embedded into the constitutive laws of the cohesive zone model (CZM) by the user-defined material subroutine. The failure prediction method was verified by the single lap joint experiment and numerical analysis before and after aging.

An Experimental Study of Granular Material Using Recycled Concrete Waste for Pavement Roadbed Construction

Rapid worldwide urbanization and drastic population growth have increased the demand for new road construction, which will cause a substantial amount of natural resources such as aggregates to be consumed. The use of recycled concrete aggregate could be one of the possible ways to offset the aggregate shortage problem and reduce environmental pollution. This paper reports an experimental study of unbound granular material using recycled concrete aggregate for pavement subbase construction. Five percentages of recycled concrete aggregate obtained from two different sources with an originally designed compressive strength of 20–30 MPa as well as 31–40 MPa at three particle size levels, i.e., coarse, fine, and extra fine, were tested for their properties, i.e., the optimum moisture content density, Californian bearing ratio, and resilient modulus. A characterization of the resilient modulus of the mixes under complex stress conditions was performed. The characterized modulus model was used in the nonlinear analysis of the pavement structure under traffic loading using KENALYER software. Consequently, the two critical responses, i.e., the tensile strain at the bottom of the asphalt layer and the vertical compressive strain at the top of the subgrade, were computed and compared for the pavement structures with varying types and percentages of recycled concrete aggregate used in the subbase layer.

Acoustic emission characteristics and failure mode analysis of rock failure under complex stress state

To study the acoustic characteristics and failure mode of the rock failure process under a complex stress state, acoustic emission (AE) monitoring technology was employed to collect the AE signals in the Brazilian indirect tension test (BITT), uniaxial compression test (UCT), and triaxial compression test (TCT) of yellow sandstone. Moreover, the evolution of AE parameters as well as the characteristics of failure mode during the failure process were analyzed. The results reveal that the peak value of AE b decreases to its lowest point before the peak stress, and the decrease in b corresponds to the sudden increase in AE energy. The peak frequency is primarily distributed in the 0–100 kHz and 200–300 kHz frequency band in BITT. It is primarily distributed in 0–100 kHz and 200–300 kHz band for UCT. The 100–200 kHz band dominates in TCT. The parameter analysis of RA (rise time/amplitude) and AF (AE counts/duration) values for classifying different cracking modes during loading reveal that the highest proportion of tensile events in BITT occurs in the failure stage near the peak stress. Tensile events dominate before the peak stress of UCT, and the proportion of shear events will increase in peak and post-peak stress. The peak value of the proportion of shear events in TCT appears in the range near the peak stress. Moreover, the classification results of the cracking modes coincide with the macrocracks. The research results can provide some reference for the study of the failure process of complex stress rock.

Геотехническое обоснование первоочередной разрезки залежей богатых руд шахты «Глубокая» методами пошагового численного моделирования в условиях гравитационно-тектонического поля напряжений

With regard to mining at depths the issues of ensuring the chambers stability, the choice of their orientation and the support selection still at the design stage are particularly relevant. At depths, near large tectonic disturbances, the factor of natural rock pressure starts to play a defining role in the underground structures stability, and the constant growth of requirements for improving safety of underground structures predetermines the need to incorporate modern approaches to identify the stressstrain state using numerical simulation. Making design decisions without simulation can lead to a decrease in economic performance and production safety, and sometimes entail the impossibility of full recovery of reserves. The finite element simulation used in this work as one of the main methods for determining the optimal option for primary cutting of the ore body and mining a rich ore body located in close proximity to large-scale tectonic disturbance that provokes a complex stress state of the ore body itself. On a practical level, the finite element method can be discredited, and therefore a description of the sequence of techniques for constructing and calibrating finite element models of primary cutting of the ore body (mining options) is required for correct identification of the most suitable mining option, that is done in this work. The integrated approach includes: formation of complex spatial geometric model of ore bodies, taking into account the morphology of bedding; separation of the model into stages of development and forming different variants of development; determination of borderline conditions for Cauchy tensor correct task; determination of deformation model and assignment of physical and mechanical parameters; carrying out verification calculations; calculation of all model variants, their analysis and selection of the most optimal ore deposit mining technique; interpolation of data into geomechanical block model for the possibility of selecting and substantiating the supports parameters and calculating the stability of chambers using analytical tools.