This paper presents a methodology for constructing a kinematic-force model of interaction of a composite elastic spherical body with two flat surfaces inclined to each other. Such problems are relevant in contact interaction mechanics because they have wide applications in engineering practice, including transport systems, robotics, aerospace engineering, and building structures. The developed approach allows reducing the complex contact problem associated with inhomogeneous distribution of kinematic and force parameters to a series of simpler model problems for which numerical-analytical solution methods have been previously developed. Special attention is paid to accounting for the effects of combined dry friction occurring within the contact spots, as well as the influence of anisotropic distributions of contact stresses. The methodology is based on numerical-analytical calculation of the stress state of a composite elastic body moving along rough surfaces under complex kinematics, including simultaneous rolling and sliding. To adequately describe the force interaction of the body with contact surfaces, approximation models are proposed that take into account the distribution of normal and tangential stresses in the contact spots. This allowed not only simplifying the mathematical description of the system but also obtaining analytical dependencies suitable for use in engineering calculations. The developed approach takes into account the key features of the interaction of an elastic body with inclined surfaces, which is important when modeling the dynamics of contact systems operating under complex friction conditions. The methodology can be applied in the analysis of mechanical systems with multipoint contacts, prediction of the dynamics of elastic spherical shells, optimization of the geometry and materials of contact elements, as well as in the development of motion control algorithms for objects supported by rough and inclined surfaces. In particular, the proposed model can be used in the design of robotic systems with complex contact interaction elements, analysis of the stability of transport and aerospace structures, as well as in the study of mechanical properties of composite materials subjected to complex loads.

An engineering method for analyzing the fragmentation of a thin spherical shell during its rapid expansion, which can occur, for example, during a strong internal explosion, is proposed. To describe the mechanical properties of the shell material, an ideally rigid-plastic model with an incompressibility condition is used. The mechanical properties of the material are characterized by two parameters: yield strength and density. The geometric parameters of the shell are determined by the radius and thickness. The problem of motion of a spherical segment with a given velocity distribution is considered. Based on the solution of this problem, an expression for the loss of kinetic energy is obtained. To estimate the energy required to form a fracture surface in the shell, the solution of the problem of fracture of an ideally rigid-plastic rod in a plane formulation is used. This allowed estimating the energy per unit surface, which is spent on fracture as a function of only two parameters — the yield strength Y and the shell thickness h. The hypothesis is adopted that the energy spent on fracture is equally distributed between neighboring fragments. From this condition, the number of fragments is estimated. A simple engineering estimate of the number of fragments and the mass of an individual fragment is obtained under the simplest assumption that all fragments are equal. An engineering method for estimating such parameters is proposed, which allows obtaining a non-uniform distribution of fragments by mass. Calculations of sphere fragmentation taking into account the non-uniform distribution of fragments by mass are carried out and presented, along with their comparison with estimates when the fragments were assumed to be identical.

The problem of natural vibrations of a five-layer rod symmetric in thickness with two cores is considered. The load-bearing layers are assumed to be thin and high-strength. For them, Bernoulli’s hypotheses about cross-sections being flat and perpendicular to the deformed axis after load application are adopted. In the relatively thick lightweight cores, the Timoshenko hypothesis is fulfilled, according to which the cross-section remains flat and incompressible but rotates by some additional angle. The thickness of the load-bearing layers is ten times less than the thickness of the cores. The system of differential equations describing the natural vibrations of the rod is obtained by the variational method. The arising transverse inertia forces are taken into account. For a thickness-symmetric rod, the system reduces to two partial differential equations for the deflection and relative shear in the cores. An algebraic equation for determining the eigenvalues is given. The first 14 of them and the corresponding natural vibration frequencies for a rod with rigidly clamped ends are calculated. Their values are summarized in a table. The analytical solution of the initial-boundary value problem on natural vibrations of a five-layer thickness-symmetric rod is obtained by expanding the desired deflection and relative shear into a series using the constructed system of eigenfunctions. The dependence of the first three frequencies on the thickness of the central load-bearing layer is numerically investigated. Graphs of the fundamental frequency dependence on the thickness and materials of the outer load-bearing layers and cores are presented. Duralumin, titanium alloy, cordierite were taken as the materials of the load-bearing layers. Cores: fluoroplastic-4, polyurethane foam, foam plastic.

The article considers the solution of the problem of shear stress distribution in a rod of variable cross-section within the framework of gradient elasticity theory. A rod is understood as a body to which only tensile or compressive axial loads are applied. The aim of the work is to refine the distribution of shear stresses along the rod length. Shear stresses along the longitudinal axis of the rod can affect delamination of layers in composite structures. The formulation of the rod theory is carried out on the basis of the variational approach. From the variational formulation, equilibrium equations for the rod and boundary conditions are obtained. By integrating the equilibrium equations and using the resulting boundary conditions, longitudinal displacements can be found. From the displacement function, solutions for normal and shear stresses can be obtained. The general formulation of the problem for different cross-sectional dimensions is considered. A test analytical solution for a linear function of the change in the cross-sectional radius of the rod is constructed. The solution is constructed for a rod consisting of two sections. The first section has a constant cross-section. In the second section, the cross-sectional radius depends linearly on the longitudinal coordinate. Graphs of normal and shear stresses for various values of additional non-classical constants at a constant value of the classical Young’s modulus are presented. A comparison is shown with the solution constructed on the basis of classical linear elasticity theory and a numerical three-dimensional solution constructed by the finite element method. It is shown that the resulting solution is continuous. It better corresponds to the numerical three-dimensional solution and allows ensuring the continuity of shear stresses along the rod axis. Recommendations for choosing the scale parameter of the theory are proposed.

Working elements made of shape memory alloy (SMA) for actuators that create large displacements are conveniently chosen in the form of beams operating in bending. When designing such actuators, it is necessary to consider the possibility of accumulation of irreversible microplastic deformation and the shift in temperatures at which deformation recovery occurs due to the martensite stabilization effect (MSE). Modeling of the operation of such an element is performed in this work. Elements in the form of a beam made of Ti50Ni50 SMA and a composite beam containing two layers: one of Ti50Ni50 and the other of Ti49.3Ni50.7 are considered. In numerical experiments, preliminary deformation was specified at a temperature at which the Ti50Ni50 alloy is in the martensitic state and the Ti49.3Ni50.7 alloy is in the austenitic pseudoelastic state. Constitutive relations were specified within the framework of the microstructural model taking into account microplasticity and MSE. Bending was considered within the Bernoulli scheme. The boundary value problem of mechanics was solved by reducing to the problem of a fixed point of an operator equal to the composition of the operator calculating the stress field and the operator expressing the increment of inelastic deformation. Bending diagrams in the form of dependencies of the bending moment on beam deflection and beam deflection on temperature during heating are found. The distributions of stresses and martensite volume fraction along the beam height at various stages of preliminary deformation and subsequent heating are calculated. It is shown that MSE under conditions of non-uniform preliminary deformation leads during subsequent heating and deflection recovery to a non-uniform occurrence of the reverse martensitic transformation and to a complex stress distribution along the beam thickness. Neglecting MSE leads to significant errors in estimating the shape recovery temperatures of the beam.

As is known, carbon nanotubes (nanofibers) in any environment (solution, suspension, melt, solid phase) form ring-like formations structurally analogous to macromolecular coils of branched polymer chains. This analogy is confirmed in this work, where the proposed model considers nanotubes (nanofibers) as an analog of a macromolecular coil, and the polymer matrix as an analog of a solvent. Structurally, these nanofillers are interpreted within the framework of the freely jointed chain model, widely used to describe the behavior of polymer macromolecules. This approach allows applying the methods of fractal physical chemistry of polymer solutions for the same purposes, since both macromolecular coils and ring-like formations of carbon nanotubes (nanofibers) are fractal objects. Comparison of the size (radius) of ring-like formations calculated according to percolation theory and the freely jointed chain model showed their good qualitative and satisfactory quantitative agreement. As expected, the much higher radius of these formations compared to the macromolecular coil is due to the high molecular weight of nanotubes (nanofibers). The rather significant discrepancy in the absolute values of the radius of ring-like formations of nanotubes (nanofibers) calculated according to these two models (about 13%) is explained by the variation of the fractal dimension of ring-like formations within 1.75-2.29, which differs significantly from the dimension of the Gaussian macromolecular coil postulated in the freely jointed chain model, whose dimension is 2. Constructing the dependence of the discrepancy in the radii of ring-like formations on their fractal dimension showed that at dimension 2 (Gaussian macromolecular coil), this discrepancy tends to zero, i.e., these models give a consistent result. This means that the percolation model allows obtaining a more accurate estimate of the radius of ring-like formations due to the absence of conformational restrictions imposed on it.

When identifying the parameters of deformation-type constitutive relations for cohesionless soils, it is necessary to have a set of experiments from which the dependencies of the first two invariants of stress tensors on similar invariants of strain tensors can be determined. Standard experimental geotechnical equipment does not allow performing the necessary measurements. For example, based on the results of experiments on such instruments, it is impossible to determine the transverse strain coefficient (Poisson’s ratio) of the soil. In compression tests, the dependence of vertical stress on uniaxial strain is measured, but the resulting radial stress in the soil cannot be measured, and therefore it is impossible to calculate Poisson’s ratio. Literature analysis shows that Poisson’s ratio values given by various authors often differ from those recommended by regulatory documents. Poisson’s ratio of dispersed soils is determined during axisymmetric tests in a triaxial apparatus. However, in this experiment, a significantly non-uniform stress-strain state is created: an uneven distribution of horizontal deformations along the sample height occurs, and the sample takes the shape of a barrel with an uneven surface. This affects the accuracy of determining the soil Poisson’s ratio in the triaxial apparatus. In our work, a method for determining the transverse strain coefficient of soil is proposed, which is a modification of the quasi-static compression experiment. Instead of an absolutely rigid cage that does not allow the sample to deform in the transverse direction, a thin deformable shell with known elastic properties is used. When calculating the transverse strain coefficient, the values of the shell circumferential strain, vertical stress and vertical strain measured during the experiment, as well as the elastic properties and geometric dimensions of the shell, are used. The experiment is implemented on a standard uniaxial testing machine. Experiments are conducted for two types of quartz sands of different particle sizes. The experimental results are verified by comparison with measurements in compression tests. The obtained Poisson’s ratio values correspond to the tendency of Poisson’s ratio change with increasing volumetric deformation known from the literature.

The results of studies of elastic and strength characteristics of three-layer panel samples under edge compression, cut from a real civil aircraft structure, consisting of thin carbon fiber skins and aluminum honeycomb core made from Russian materials, are presented. It is established that when testing three-layer panels for edge compression, to obtain accurate characteristics of material properties with the smallest scatter of values, as well as to avoid invalid failure in the form of crushing in the load application zone, it is necessary to strengthen the loaded ends. A computational and experimental substantiation of the design parameters of the samples is performed, in particular, rational values of the depth of end strengthening and sample width are obtained to ensure failure in the working, valid zone, as well as to minimize labor intensity during their preparation for strength testing under edge compression. A method for preparing samples for testing is developed taking into account the requirements of GOST R 56809-2015, and an adhesive compound that can be used for end strengthening is selected. An analysis of the applicability of the strengthening method for a sample of standard geometry (in accordance with the GOST and ASTM test methods) for different core thicknesses is carried out to determine the limitations of this method in edge compression testing and to obtain sample failure in the working, valid zone. In addition, an assessment of the thickness limitations of three-layer panels for the selected strengthening method is carried out using finite element calculation. The permissible thickness of the three-layer panel at which failure in the valid zone is ensured is obtained; for panels of greater thickness, significant plastic deformation accumulates in the core, which ultimately leads to invalid failure.

The article is devoted to the development of an experimental stand designed to study the dynamics of settlement of polymer materials (including polymer composite materials) under the influence of high-intensity ultrasonic vibrations. The operation of the experimental stand is based on monitoring the displacement of the ultrasonic emitter, which, during contact interaction with the test sample, under static pressure by which the ultrasonic emitter is pressed against the test sample, becomes movable as the sample absorbs ultrasonic energy and, as a result, transitions of the sample from solid to viscous and then to fluid state. The experimental data obtained using the developed stand, characterizing the settlement dynamics of polymer materials together with other data characterizing the ultrasonic impact process (ultrasonic frequency, power introduced into the ultrasonic zone, the nature of its change, etc.) will allow a more detailed study of the stages of the process of ultrasound action on the polymer, to identify, in particular, the contribution of hydrodynamic effects (cavitation phenomena, acoustic microflows, viscosity relaxation effects) to the change in properties and intensification of diffusion at the polymer interface under the action of powerful ultrasonic vibrations. Experimental data are obtained in the range of ultrasonic vibration amplitudes from 5 to 40 μm and a range of static forces up to 180 N. The range of controlled deformations is 5 mm. The developed stand will allow studying and fully revealing the potential of ultrasonic welding of polymer materials, in particular, it will allow developing methods for effective implementation of the welding process for both homogeneous and dissimilar materials.

The paper presents the results of numerical simulation of the destruction process of metal structures containing ice under the action of an ogival striker. In the structure (simulating passive protection), thin metal plates were located above and below the ice, as if protecting and simultaneously supporting it. The number of plates under the ice was increased until through penetration became impossible. Ice without phase transitions with averaged physical and mechanical characteristics was considered. The behavior of ice was modeled by the relations of continuum mechanics based on a phenomenological macroscopic approach. The concept of material failure based on a deterministic approach was used. The G.R. Johnson numerical method was used as the main tool, the computational part of which is supplemented by mechanisms of computational node splitting and computational element destruction. Numerical simulation was carried out in an axisymmetric formulation using the non-commercial software package “Impact. Axisymmetric Problem”. Qualitative and quantitative tests were performed, and a finite element mesh sensitivity analysis was made. The calculation results were obtained in the form of final configurations of the “striker-target”, graphs and tables. The main regularities of the penetration process (the time of initiation of the first failure sites and their evolution, the mechanism of structure penetration and ice destruction) were studied, and the level of residual deformation of the plates and the process time were qualitatively and quantitatively assessed. It was revealed that the brittle mechanism dominated in the process of ice destruction, and the penetration of structures occurred by the “puncture” mechanism. After penetration of the upper layers, only elastic deformation of the striker was noted, and after penetration of the lower layers, its plastic deformation with blunting and fragmentation of the tip was noted. It was established that the presence of 11 steel plates of 1 mm thickness under the ice block led to a complete stop of the striker.