Contact Plane Research Articles

Tensor based approach for tooth contact analysis of planar and spatial gearing contact

This paper presents a new method for analyzing gear tooth contact, leveraging tensor notation and discrete differential geometry. Existing analytical and numerical methods often face challenges in convergence and computational efficiency. Our proposed approach involves defining a novel tensor notation and applying it to the gear tooth contact kinematics problem. By discretizing the tooth surfaces and using tensor operations, we can accurately determine the kinematics, path of contact, and transmission error of the gear pair. To validate our approach, we compared its results with those obtained from commercial software, KISSsoft and KiMOS. The results demonstrated high accuracy, with mean absolute errors below 0.08 for specific sliding, 0.003 for sliding factor, and 0.1μm for transmission error. Furthermore, we applied our method to analyze non-involute gear types, such as S-gears, pin gearing and cosine gears. Our findings revealed the kinematic performance and contact characteristics of these gear types, providing valuable insights for gear design and optimization. In conclusion, the proposed tensor-based approach offers a promising alternative for gear tooth contact analysis, providing accurate and efficient results for a wide range of gear types.

Development Characteristics of Single Sand Body of Yan’An Formation in Western Ordos Basin

Abstract The precise characterization of the types and spatial distribution of strongly heterogeneous single sand bodies are a necessary prerequisite for the study of oil and gas reservoir formation. This article takes the Yan’an Formation in the western Ordos Basin as an example to systematically study the internal structure and profile distribution of strongly heterogeneous single sand bodies. The research results indicate that using mud interlayers, physical interlayers, and calcium interlayers as layered interfaces can effectively classify the single sand body types of each small layer in the Yan’an Formation. The drilling rate of a single sand body is between 50% and 85%, and the average number of sand layers is between 3 and 8. When two rivers intersect and are horizontally tangent, due to the mutual erosion of water flow, relatively thick sand bodies can be retained, ultimately resulting in a pattern of relatively thick on both sides and thin in the middle. The single sand body at the bottom is the most developed and has good continuity. In the transverse direction of the river, the continuity of a single sand body is relatively poor, and the extension range of a single sand body is 300m to 1200m. Along the river channel, single sand bodies are developed with good continuity, extending about 2km. The connectivity between the sand body and the well varies with the migration of the river in the transverse river profile of the research area. The lateral contact relationship of a single sand body determines the boundary of the river channel, and the profile characteristics of a single sand body also reflect its planar distribution. The intergranular pores of the Yan’an Formation sand body are relatively developed, and a large number of quartz particles offset the compressive stress of the rock. The corrosion of feldspar particles is relatively strong, retaining a large number of primary intergranular pores. There is a good quantitative relationship between the thickness of sand in a single level channel and the width of the channel. The control of the well network on the sand body is relatively small, and there is room for further densification of drilling. The vertical contact relationship of a single sand body in the Yan’an Formation of the research area can be divided into three modes: separation, stacking, and overlapping. Cut pile sand bodies with good connectivity have good production capacity. The different planar contact relationships reflect the degree of connectivity between channel sand bodies and wells, with lateral cutting type>docking type>isolation type. The side cut sand body has uniform longitudinal water absorption, small water absorption difference, and high-water drive degree. The isolation type has the greatest difference in water absorption rate and the lowest degree of water flooding.

Semi-empirical terramechanics modelling of rough terrain represented by a height field

Dynamic simulations of various types of off-road vehicles, from planetary rovers to agricultural equipment, have long relied on well-established semi-empirical terramechanics models. While these models do have drawbacks and reliability issues that have been addressed by numerous works in the decades since the models were first introduced, semi-empirical approaches remain one of the few ways to simulate realistic wheel-soil interaction in real-time. One of their drawbacks is their assumption that the terrain is a flat plane. The models work by integrating normal and shear stresses along the wheel-terrain contact patch. The normal stress at each point along the contact patch is determined using an equation that computes soil pressure based on semi-empirical parameters, the dimensions of the wheel and the sinkage, which is determined based on the distance between the point and the plane that defines the terrain. Other works simplify the rough terrain contact problem by defining an equivalent contact plane at each time step in order to continue to be able to use semi-empirical models - modified to work with slanted planes - to compute the interaction forces. In this work, we propose a new, modified version of the semi-empirical model in which interaction forces for a wheel travelling on rough terrain can be computed without the need to use an equivalent contact plane. To highlight the advantages of our proposed approach, we compare our simulation results to the results of simulations using an existing approach for modelling a wheel travelling over rough terrain using traditional semi-empirical models.

Modelling of 2-D seismic wave propagation in heterogeneous porous media: a frequency-domain finite-element method formulated by variational principles

SUMMARY We propose a frequency-domain finite-element (FDFE) method to model the 2-D P–SV waves propagating in porous media. This specific finite-element method (FEM) is based on the framework of variational principles, which differs from previously widely used FEMs that rely on the weak formulations of the governing equations. By applying the calculus of variations, we establish the equivalence between solving the stress–strain relations, equations of motion and boundary conditions that govern the propagation of P–SV waves, and determining the extremum or stationarity of a properly defined functional. The structured rectangular element is utilized to partition the entire computational region. We validate the FDFE method by conducting a comparison with an analytically-based method for models of a horizontal planar contact of two poroelastic half-spaces, and a poroelastic half-space with a free surface. The excellent agreements between the analytically-based solutions and the FDFE solutions indicate the effectiveness of the FDFE method in modelling the poroelastic waves. Modelling results manifest that both propagative and diffusive natures of the Biot slow P wave can be effectively modelled. The proposed FDFE method simulates wavefields in the frequency domain, allowing for easy incorporation of frequency-dependent parameters and enabling parallel computational capabilities at each frequency point (sample). We further employ the developed FDFE method to model two simplified poroelastic reservoirs, one with gas-saturated sandstone and the other with oil-saturated sandstone. The results suggest that changing the fluid phase of the sandstone reservoir from gas to oil can substantially impact the recorded solid and relative fluid–solid displacements. The modelling suggests that the proposed FDFE algorithm is a useful tool for studying the propagation of poroelastic waves.

Observation of voltage dependent negative differential resistance (NDR) in SnS2-GO nanocomposites

This study investigates the structural, optical, and electrical properties of tin disulfide (SnS2) and SnS2-graphene oxide (GO) nanosheets synthesized via chemical bath deposition method (CBD). Structural characterization confirms the formation of hexagonal crystal phases with nanosheet morphology. It shows a well distribution of nanosheet average square sizes of 10 nm for SnS2 and 6 nm for SnS2-GO. Optical analysis shows blue shifts in absorption edges compared to bulk SnS2, attributed to quantum confinement effects. Photoluminescence emission peaks exhibit different energy levels in SnS2-GO originated to native defects. The composites show a sharp reduced of PL intensity due to enhanced charge carrier separation. Electrical measurements on SnS2-GO thin films demonstrate negative differential resistance (NDR) behavior in both planar and sandwich contact configurations, suggesting electron injection/extraction mechanisms. The NDR phenomenon exhibits a dependence on voltage scan rate, indicating the involvement of electronic and ionic elements in charge transport mechanisms. Overall, this study provides insights into the NDR properties of SnS2-GO nanocomposite, laying the groundwork for their potential applications in optoelectronics and nanoelectronics.

Experimental assessment of strength characteristics of anthropogenic soils in a given mining area for justification of tractor propulsion type

BACKGROUND: In mining, there is a need to develop tractors as special robotic vehicles of the robotic mining facility that can efficiently move and work on soils with low bearing capacity, in waterlogged areas and in the underwater position. A preliminary evaluation of the strength characteristics of such a heterogeneous surface of the exploitation area helps to choose an effective concept and technical characteristics of the propulsor of the tractor. AIM: Formation of the database of the physical condition and mechanical properties of the environment in which the tractor will move and operate. METHODS: In order to achieve the aim, preliminary experiments were conducted on the terrain. Typical kinds of soil sites and watered areas, including those of anthropogenic origin, of a given mining area were identified. Full-scale instrumental tests were conducted to determine: granulometric composition of soils; physical and mechanical parameters of these soils at the selected sites using the developed original types of a penetrometer and a shear rig. RESULTS: The database of numerical values of measured strength characteristics of one of the soil types (as an example) as a possible supporting surface for vehicle motion in the area of mining facility operation is presented in a graphical form. The basis of this database is the graphs according to the results of experiments on free settlement and shear with the help of dies of two types of configuration (the round die and the original caterpillar track respectively); the granulometric composition of the selected soils has been evaluated. CONCLUSION: The methodology of preliminary assessment of the soil condition by two indicators of soil strength (the rut depth z depending on the average pressure under the die) and shear (average tangential stresses τ in the contact patch on the motion of the die relative to the soil, depending on the ratio of the normal force and the traction force in the contact plane) has been developed. The 3D model of the coastal and bottom area of the technogenic mining zone has been developed. Design variants of highly efficient types of tractor propulsors are selected on the basis of prediction of maximum allowable stresses in the contact patch with the ground for the bearing surface areas of the considered area of operation.

Heterotypic interaction promotes asymmetric division of human hematopoietic progenitors.

Hematopoietic stem and progenitor cells (HSPCs) give rise to all cell types of the hematopoietic system through various processes, including asymmetric divisions. However, the contribution of stromal cells of the hematopoietic niches in the control of HSPC asymmetric divisions remains unknown. Using polyacrylamide microwells as minimalist niches, we show that specific heterotypic interactions with osteoblast and endothelial cells promote asymmetric divisions of human HSPCs. Upon interaction, HSPCs polarize in interphase with the centrosome, the Golgi apparatus, and lysosomes positioned close to the site of contact. Subsequently, during mitosis, HSPCs orient their spindle perpendicular to the plane of contact. This division mode gives rise to siblings with unequal amounts of lysosomes and of the differentiation marker CD34. Such asymmetric inheritance generates heterogeneity in the progeny, which is likely to contribute to the plasticity of the early steps of hematopoiesis.

Cyclicity of katatectic layers in anhydrite caprocks, U.S. Gulf Coastal Region

ABSTRACT An anhydrite caprock overlies 65% of the 329 onshore salt stocks of the U.S. Gulf Coastal Region, south-central U.S. The caprock consists of a succession of katatectic (downward-building) layers of anhydrite (typically 1–4 cm thick) each overlain by an upper, black pyrite lamina (∼ 0.1 mm thick). The older (upper) katatectic layers are usually highly deformed. During four stages, these fundamental layers formed at the crest of shallow (< 1,200 m) salt stocks. Repetition of sets of the stages resulted in the cyclicity. The closed stage A flush, nearly planar contact separated mature anhydrite caprock from an underlying salt-stock crest. As the diapir elevated by a few mm/yr, it arched the caprock; it also opened conduits at the stock’s uppermost anhydrite sheath, allowing NaCl-undersaturated water from outside the diapir to contact the salt stock. The open-dynamic stage: Along the perimeter of the uppermost salt stock, a narrow, annular halite cave advanced inward and upward directly below the slightly convex base of the anhydrite caprock. The most NaCl-aggressive water rose by free convection to the highest elevations and dissolved halite. The dense, solute-enhanced water then reversed direction and, again, by free convection, flowed past the stock’s margin into country rock; NaCl-undersaturated water replaced the departing brine. A water-filled, commonly ring-shaped, planar cave up to several meters high expanded over the stock’s crest. Sparse physical supports, buoyancy, arched caprock, and high artesian water pressure prevented the cave’s collapse. The cave’s halite floor dissolved vertically downward faster than it moved diapirically upward. As the crestal halite dissolved, a small amount of residual pyrite and, typically, a layer up to ∼ 10 cm thick of residual anhydrite (generally 3–8 wt % of the stocks) accumulated on the cave floor. The open-static stage: The persistent “attack” of halite highs by the NaCl-undersaturated water eventually caused the slope of the cave’s floor to approach horizontality. The density-driven flow slowed markedly, and a static cover of NaCl-saturated brine thwarted the dissolution of the halite floor. The downward movement of the cave’s floor reversed, and it moved slowly upward via diapirism. Concurrently, an anhydrite sheath evolved around the uppermost stock. The accretionary stage The cave closed in a few thousand years, and active diapirism underplated the admixed residual minerals onto the base of the caprock. The uppermost, now compact, residual anhydrite dissolved in the high-pressure environment, leaving the discrete, topmost, thin, black pyrite lamina. With the accretion of the residual minerals, a katatectic layer formed at the base of a succession of such layers. The closed stage recurred, and a new katatectic cycle began forming. Variations of the four genetic stages probably occurred during the formation of anhydrite caprocks in other worldwide salt-dome provinces, but because of insufficient data, they are usually unrecognizable.

Pressure-Angle Polars Obtaining Flow Field of Rotating Detonation

ABSTRACT Rotating detonation engines have been studied experimentally and numerically due to a higher thermal efficiency in theory. Numerical simulations of the flow field inside the engine cost more since the flow is unsteady and contains shock waves, detonation waves, and contact surfaces. Thus, a two-dimensional theoretical study with the method of pressure behind a shock wave or a rarefaction wave as a function of deflection angle of flow was carried out to predict the flow field and reduce the cost. Numerical simulations with a premixed inviscid model were also carried out to verify the theoretical results. The flow field at an equivalence ratio of 0.8 has a curved contact surface, whereas it has a plane contact surface at equivalence ratios of 0.9–1.2 because the RDW is stronger and more stable. Comparisons between the theoretical and numerical results show the theoretical method can be used to obtain a rough flow field inside the RDE, including the pressure and deflection angle of flow behind the shock wave and rarefaction wave. Most of the errors between the numerical and theoretical results are less than 10%.

Прогнозирование состояния плоскостных контактных соединений шин с применением моделей машинного обучения

The article considers the problem of predicting the technical condition of planar contact connections of tires in electrical installations. Modeling methods, including polynomial regression and autoregression, have been studied to assess and predict the state of contacts. The analysis of the results of the models was carried out and their effectiveness was evaluated on various data sets. A program for predicting the condition of contacts has been developed and integrated into the software for monitoring tire contacts. The results of the work show the prospects of creating monitoring systems using forecasting models to ensure the reliability and efficiency of electrical equipment. Keywords: MONITORING, CONTROL, TECHNICAL CONDITION, ELECTRICAL INSTALLATION, CONTACT CONNECTIONS

Non-adhesive and adhesive contacts of an elastic quarter-or eighth-space with freely sliding sides

The contact of an elastic quarter- or eighth-space is studied under the condition that the movement of the side surface of the quarter-space is constrained: It can slide freely along the plane of the side surface but its normal movement is blocked (for example, by a rigid wall). The solution of this contact problem can be easily achieved by additionally applying a mirrored load to an elastic half-space. Non-adhesive contact and the Johnson-Kendall-Roberts (JKR)-type adhesive contact between a rigid sphere and an elastic quarter-space under such a boundary condition is numerically simulated using the fast Fourier transform (FFT)-assisted boundary element method (BEM). Contacts of an elastic eighth-space are investigated using the same idea. Depending on the position of the sphere relative to the side edge, different contact behavior is observed. In the case of adhesive contact, the force of adhesion first increases with increasing the distance from the edge of the quarter-space, achieves a maximum, and decreases further to the JKR-value in large distance from the edge. The enhancement of the force of adhesion compared to the half-space-contact is associated with the pinning of the contact area at the edge. We provide the maps of the force of adhesion and their analytical approximations, as well as pressure distributions in the contact plane and inside the quarter-/eighth-space.

Nanoskiving of van der Waals Materials toward Edge/Basal Plane Contact Heterojunctions for High-Performance Photodetection.

The unique features of edges in van der Waals materials may lead to edge-basal plane contacts that could provide new opportunities for electronic and optoelectronic devices. However, few studies have addressed edge/basal plane contact heterojunctions owing to the formidable challenges in integrating edges with the basal plane to form a heterojunction. Here, taking the example of black phosphorus (BP)/ReS2, a heterojunction with contact between the edge and the basal plane was successfully achieved by the introduction of a nanoskiving technique to fabricate BP edges with controlled orientation, followed by the dry transfer of a ReS2 flake. The deformation of BP during the nanoskiving process was clearly revealed, where interlayer slipping in the BP determined the formation of the edges. The edge/basal plane contact heterojunctions based on BP/ReS2 exhibited a reverse-rectifying behavior upon contact, and a high rectifying current was attributed to direct tunneling and Fowler-Nordheim tunneling in low and high bias regimes, respectively. As a photodetector, the heterojunction diode demonstrated an impressive responsivity of 65 A/W, a rapid response time (<10 ms), and polarization-sensitive detection under 532 nm illumination without gate biasing.

The robust superlubricity behaviors of a-C:H films on different roughness interfaces at surface contact scale

Hydrogenated amorphous carbon (a-C:H) film has demonstrated superlubricity in previous studies, but its effectiveness may be limited when applied to engineering components with surface contact and non-ideal smoothness. In this work, four kinds of plane contact friction pairs with different surface roughness (roughness of Sa from 21 nm to 544 nm) were designed and coated with a-C:H film with 43% hydrogen content by ionized beam deposition method. The results demonstrate that the film exhibits exceptional lubrication performance across various roughness interfaces, with the minimum friction coefficient reaching 0.005. The graphitization process and the actual contact state are discussed and calculated. These findings offer valuable insights into the potential application of highly-hydrogenated a-C:H film for moving parts in industrial equipment.

A transmission electron microscopy investigation suggests that telocytes, skeletal muscles, myoblasts, and stem cells in common carp (Cyprinus carpio) respond to salinity challenges

BackgroundTelocytes are modified interstitial cells that communicate with other types of cells, including stem cells. Stemness properties render them more susceptible to environmental conditions. The current morphological investigation examined the reactions of telocytes to salt stress in relation to stem cells and myoblasts. The common carp are subjected to salinity levels of 0.2, 6, and 10 ppt. The gill samples were preserved and prepared for TEM.ResultsThe present study observed that telocytes undergo morphological change and exhibit enhanced secretory activities in response to changes in salinity. TEM can identify typical telocytes. This research gives evidence for the communication of telocytes with stem cells, myoblasts, and skeletal muscles. Telocytes surround stem cells. Telopodes made planar contact with the cell membrane of the stem cell. Telocytes and their telopodes surrounded the skeletal myoblast. These findings show that telocytes may act as nurse cells for skeletal stem cells and myoblasts, which undergo fibrillogenesis. Not only telocytes undergo morphological alternations, but also skeletal muscles become hypertrophied, which receive telocyte secretory vesicles in intercellular compartments.ConclusionIn conclusion, the activation of telocytes is what causes stress adaptation. They might act as important players in intercellular communication between cells. It is also possible that reciprocal interaction occurs between telocytes and other cells to adapt to changing environmental conditions.

Effect of Low-Temperature Plasma Carburization on Fretting Wear Behavior of AISI 316L Stainless Steel

AISI 316L stainless steel has received considerable attention as a common material for key ball valve components; however, its properties cannot be improved through traditional phase transformation, and fretting wears the contact interface between valve parts. A carburized layer was prepared on the surface of AISI 316L stainless steel by using double-glow low-temperature plasma carburization technology. This study reveals the effect of double-glow low-temperature plasma carburization technology on the fretting wear mechanism of AISI 316L steel under different normal loads and displacements. The fretting wear behavior and energy dissipation of the AISI 316L steel and the carburized layer were studied on an SRV-V fretting friction and wear machine with ball–plane contact. The wear mark morphology was analyzed by using scanning electron microscopy (SEM), the phase structure of the carburized layer was characterized with X-ray diffractometry (XRD), and the wear profile and wear volume were evaluated with laser confocal microscopy. The carburized layer contains a single Sc phase, a uniform and dense structure, and a metallurgically combined matrix. After plasma carburizing, the sample exhibited a maximum surface hardness of 897 ± 18 HV0.2, which is approximately four times higher than that of the matrix (273 ± 33 HV0.2). Moreover, the surface roughness was approximately doubled. The wear depth, wear rate, and frictional dissipation energy coefficient of the carburized layer were significantly reduced by up to approximately an order of magnitude compared with the matrix, while the wear resistance and fretting wear stability of the carburized layer were significantly improved. Under different load conditions, the wear mechanism of the AISI 316L steel changed from adhesive wear and abrasive wear to adhesive wear, fatigue delamination, and abrasive wear. Meanwhile, the wear mechanism of the carburized layer changed from adhesive wear to adhesive wear and fatigue delamination, accompanied by a furrowing effect. Under variable displacement conditions, both the AISI 316L steel and carburized layer mainly exhibited adhesive wear and fatigue peeling. Oxygen elements accumulated in the wear marks of the AISI 316L steel and carburized layer, indicating oxidative wear. The fretting wear properties of the AISI 316L steel and carburized layer were determined using the coupled competition between mechanical factors and thermochemical factors. Low-temperature plasma carburization technology improved the stability of the fretting wear process and changed the fretting regime of the AISI 316L steel and could be considered as anti-wearing coatings of ball valves.

Single-particle fabric tensors for assemblies of spherical particles

A novel method to calculate single-particle fabric tensors and related geometric characteristics (contact, free and particle area, particle volume and contact centroid) of assemblies of spherical particles has been developed. A standard contact detection stage is used to identify overlapping particles. The shape of each contact region is obtained as the intersection of a circle, obtained in the absence of any contact impingement, and a convex polygon, determined by half-plane intersection based on directed lines obtained from the intersection of contact planes. An underlying power diagram is used to subdivide volume and area between particles, but calculations are performed particle-wise, resulting in an efficient algorithm with a structure ideally suited for parallelisation. Numerical tests performed on loose and dense particle assemblies and comparison (of single-particle free area and particle volume) with a reference method (POWERSASA) indicate that the method is robust and highly accurate. The method will be useful not only for geometric analysis but also as a basis for subsequent developments of strain measures and computational procedures for highly deformed granular materials. Moreover, it constitutes a viable alternative for calculation of solvent accessible surface areas, considering its robustness, efficiency and ease of parallelisation.

Relationship of tissue dimensions and three captive bolt application sites on cadaver heads from mature swine (Sus scrofa domesticus)

Penetrating captive bolt (PCB) is a common method of euthanasia for swine but has not been evaluated for mature swine < 200kg body weight (BW). The objectives were to determine tissue depth, brain contact plane, and visible brain tissue damage (brain damage[BD]) for the common FRONTAL (F) and alternative TEMPORAL (T) and BEHIND EAR (BE) placements for PCB use on sows and boars weighing < 200kg. Cadaver heads were obtained from 30 sows and 30 boars (estimated BW, mean ± SD; sows: 165.8 ± 22.4kg; boars: 173.6 ± 21.4kg) from a slaughter establishment after electrical stunning and exsanguination. Heads were cooled at 2 to 4 °C for approximately 64h. A Jarvis PAS-Type P 0.25R PCB with a Long Stunning Rod Nosepiece Assembly and a 3.5 GR power load was used for all PCB applications at the following placements: F-3.5cm superior to the optic orbits at midline, T-at the depression posterior to the lateral canthus of the eye within the plane between the lateral canthus and the base of the ear, or BE-directly caudal to the pinna of the ear on the same plane as the eyes and targeting the middle of the opposite eye. For sows, the bolt path was in the brain for 10/10 (100.0%, 95% CI: 69.2% to 100.0%) F, T, and BE heads. In heads that could reliably be assessed for BD, BD was detected in 10/10 (100.0%, 95% CI: 69.2% to 100.0%) F heads, 9/9 (100.0%, 95% CI: 66.4% to 100.0%) T heads, and 0/10 (0.0%, 95% CI: 0.0% to 30.1%) BE heads. For boars, the bolt path was in the plane of the brain for 8/9 (88.9%, 95% CI: 51.8% to 99.7%) F heads, 9/10 (90.0%, 95% CI: 55.5% to 99.7%) T heads, and 11/11 (100.0%, 95% CI: 71.5% to 100.0%) BE heads. In heads that could reliably be assessed for BD, BD was detected in 8/9 (88.9%, 95% CI: 51.7% to 99.7%) F heads, 7/10 (70.0%, 95% CI: 34.8% to 93.3%) T heads, and 4/11 (36.4%, 95% CI: 10.9% to 69.2%) BE heads. Tissue depth was reported as mean ± SE followed by 95% one-sided upper reference limit (URL). For sows, total tissue thickness differed (P < 0.05) between placements (F: 49.41 ± 2.74mm, URL: 70.0mm; T: 62.83 ± 1.83mm, URL: 76.6mm; BE: 84.63 ± 3.67mm; URL: 112.3mm). Total tissue thickness differed (P < 0.05) between placements for boars (F: 54.73 ± 3.23mm, URL: 77.6mm; T: 70.72 ± 3.60mm, URL: 96.3mm; BE: 92.81 ± 5.50mm; URL: 135.3mm). For swine between 120 and 200kg BW, the F placement may have the greatest likelihood for successful euthanasia due to the least total tissue thickness and may present less risk for failure than the T and BE placements.

Calculation of the decomposition coefficients for plane contact problem kernel in the orthonormal basis

Analytical solutions of some contact problems are infinite functional series according to the system of basic functions. When constructing such solutions, it becomes necessary to represent the kernels of integral equations describing the process of interaction in the form of two-dimensional series on a given basis. Often the kernels have a rather complex appearance, therefore, the process of finding the decomposition coefficients is a rather complex and labor-intensive process, on which the accuracy and speed of obtaining final results depend. The paper proposes a calculation method that allows calculating the coefficients of decomposition of the kernels of plane contact problems according to a special orthonormal basis that takes into account the features of contacting bodies. Other approximate formulas are also derived for the special case when coating characteristics are constant. Based on the received presentation, conclusions and recommendations are formulated.

О резонансах академика И.И. Воровича в контактных задачах с деформируемым штампом в сейсмологии

In this paper, for the first time, methods for obtaining resonant frequencies in plane and spatial contact prob-lems for deformable stamps are presented in a complex. Their existence was predicted by academician I.I. Vorovich. He constructed one-dimensional models of de-formable stamps consisting of a rigid stamp and a spring, illustrating the occurrence of resonances. Currently, thanks to the use of block element methods, it has become possible to investigate real contact problems with a deformable stamp. Earlier in the works of the authors, the existence of a new type of earthquakes, called starting, was established. In the course of the study, some functionals remained uncomputed, the role of which was not clear. It is shown that they serve to construct dispersion curves giving resonant frequencies predicted by academician I.I. Vorovich. A study is carried out for different types of contact problems and shows how the method of constructing dispersion equations becomes more complicated, both due to the complication of the shape of stamps and the complication of the rheologies of deformable stamps. Semi-infinite and deformable stamps of simple rheology in the form of a quarter plane are considered. In addition, a semi-infinite stamp with the rheology of Kirchhoff plates is analyzed. The obtained results provide an answer to one of the methods of constructing dispersion equations for I.I.Vorovich resonances in contact problems for different deformable stamps. The study is closely related to the identification of new precursors of seismicity, as well as to solve some problems in the theory of strength.

Some remarks on the two-dimensional contact of dissimilar elastic solids under non-slipping boundary conditions

Spence's self-similarity approach is often preferred to an incremental method for solving plane, non-slipping normal contacts of dissimilar elastic solids under progressive loading. The number of plane contact problems solved in this way so far is nevertheless very limited, since explicit expressions especially for the pressure distribution and the tangential tractions within the contact area require the solution of complicated singular integral equations. In contrast, the application of the incremental method leads to much simpler integral equations, which are derived here and applied to solve some new non-slipping contact problems including the one characterized by an initial power-law gap profile function. The incremental method is also capable to determine the so-called interface mismatch eigenstrain. In Spence's method, however, the determination of its magnitude represents a mandatory step within the solution procedure. By means of a comprehensive comparison of both methods this work provides new insights into the solution of non-slipping plane normal contact problems including explicit analytical solutions for the interface mismatch eigenstrain.