## No 3 (2024)

**Year:**2024**Articles:**9**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/420**DOI:**https://doi.org/10.15593/perm.mech/2024.3

MULTIPARAMETRIC OPTIMIZATION OF THE ROTOR BLADE DESIGN OF A HELICOPTER WITH CONTROLLED GEOMETRY

#### Abstract

In flight, the rotor blades of a helicopter create significant fluctuations and noise due to changes in the aerodynamic loads acting on them when their azimuth angle changes. Various methods are used to reduce the resulting vibrations and noise. For example, with the advent of active materials, the concept of a rotor with active twisting was proposed. The actuators integrated into the main rotor blade skin create dynamic twisting and curvature of the blade, adapted at any time to flight conditions and significantly reducing vibrations and noise, as well as improving flight characteristics. This work is devoted to the multiparametric optimization of the blade design with controlled geometry. The formulation of the problem of multiparametric optimization of a composite structure based on thermo-piezoelectric analogy is formulated. The target function is selected. The parameters of optimization of the blade design are determined and constraints for the selected parameters are formulated. A method of design of the blade structure with controlled geometry has been developed, which includes three program blocks. The first block is a mathematical model. The second block is the construction of an experiment planning matrix. The third block is to obtain a surface response and search for an extremum. The optimal parameters of the active (control) and power elements of the blade design with controlled geometry are determined. The obtained solution of the optimization problem was compared with the results of direct numerical simulation. When conducting direct numerical simulation, controlled deformations of the blade under study were calculated at different values of the control electric voltage, the problem was solved in a related formulation using the obtained geometric parameters. The results of this study can be applied in the design of structures with controlled geometry.

**PNRPU Mechanics Bulletin**. 2024;(3):5-16

ASYMPTOTIC BEHAVIOUR OF THE CRACK TIP FIELDS UNDER CREEP REGIME TAKING INTO ACCOUNT DAMAGE ACCUMULATION PROCESSES

#### Abstract

The aim of the study is to identify the asymptotic stress, creep strain rate and continuity fields behavior in the proximity of the crack tip under creep conditions, taking into account the damage accumulation process, based on the finite element analysis of the stress-strain state at the crack tip in the finite element software SIMULIA Abaqus using the UMAT procedure, which allows us to describe constitutive equations that are absent in the standard set of defining equations of the FEM complex, and incorporate the damage accumulation processes into the design scheme of the FEM software. The damage accumulation phenomenon is described using the classical Kachanov–Rabotnov model, which postulates a power law linking creep strain rates and stresses, and a power law of damage accumulation, in a coupled formulation. Finite element modeling of loading of a plate with a central horizontal and inclined crack under conditions of steady state creep is performed under the assumption of the realization of a plane stress state. It is shown that in the case of steady-state creep without taking into account the process of damage accumulation, the finite element solution clearly has the asymptotic behavior of the classical Hutchinson-Rice-Rosengren solution. With the help of the developed user procedure UMAT, the coupling of two processes is realized in the computational scheme of the finite element method: the evolution of mechanical fields and the increase of damage in the vicinity of the crack tip in accordance with the canonical Kachanov-Rabotnov damage evolution model. On the basis of the analysis of the stress field obtained by finite element computations, in the vicinity of the crack tip, taking into account the damage, a new asymptotic of stress fields near the crack tip in a plate under uniaxial tension conditions was revealed, different from the asymptotic corresponding to the Hutchinson-Rice-Rosengren solution.

**PNRPU Mechanics Bulletin**. 2024;(3):17-38

EXPERIMENTAL DETERMINATION OF CRACK-TIP FIELDS: HOLOGRAPHIC INTERFEROMETRY METHOD AND DIGITAL IMAGE CORRELATION METHOD

#### Abstract

The problems of reconstruction of the stress field at the crack tips of a system of horizontal and inclined cracks in a linear isotropic elastic medium using the results of experimental studies conducted by interference-optical methods: the method of holographic interferometry and the digital image correlation are considered. The experiments are aimed at constructing a multi-component asymptotic expansion of M. Williams with the retention of regular (nonsingular) series terms for plates weakened by two interacting cracks. To restore the coefficients of the M. Williams series the interference patterns of the absolute retardation fringes (isodromes) are used. Favre's law allows us to determine the main stresses in the vicinity of the crack tip. The strain and displacement fields were determined using the digital image correlation method for a number of cracked specimens. The experimental data obtained by two interference-optical methods was used to calculate the coefficients of the multi-point asymptotic expansion of M. Williams. A new variation of the over-deterministic method is proposed, focused on the linearized Favre's law, and allowing us to find the coefficients of the asymptotic series of M. Williams (generalized stress intensity coefficients) by means of an iterative procedure, based on the results of polarization-optical measurements. To verify the results of processing all the experimental data, a computational experiment was additionally carried out using the finite element method, which made it possible to calculate generalized stress intensity factors based on the stress fields found by the finite element method. A modification of the over-deterministic method based solely on the application of stress fields associated with the crack tip is proposed. It is shown that the generalized coefficients of the M. Williams series, determined using a full-field and computational experiment, are in good agreement.

**PNRPU Mechanics Bulletin**. 2024;(3):39-56

FEATURES OF DEFORMATION OF A BIMETALLIC PLATE

#### Abstract

The features of macroscopic localization of plastic flow during uniaxial tension of a flat bimetallic plate are discussed. The extension axis of the sample was oriented normally to the direction of rolling. The studied bimetal "low-carbon steel - stainless steel" is used in chemical engineering for the manufacture of reaction columns, autoclaves, reactors, and heat exchangers. The plastic flow curve of the bimetal after the yield point in the area of large plastic deformations is located between the curves for its components - austenitic stainless steel (AISI 304) and low-carbon steel (ASTM A414 grade A). The visualization of localized plastic deformation bands and registration of the kinetics of their movement were carried out on the working part of the sample by the method of digital speckle photography. It has been established that at the yield plateau, plastic deformation in the form of Lüders fronts originates at the interface between the cladding layer and the base bimetal layer and propagates in the base layer of low-carbon steel, while the less plastic cladding layer of stainless steel deforms elastically. Then, together with the base one, the cladding layers also begin to plastically deform in the form of the propagation of Portevin-Le Chatelier fronts. The process of failure of a bimetal also begins with the localization of plastic deformation near structural inhomogeneities and stress concentrators in the area of contact between layers of two dissimilar metals. The stress concentrators formed at the early stages of plastic flow in this region initiate the formation of a high-amplitude peak of strain localization, which is a precursor to the formation of a neck in the sample and further ductile failure of the bimetal.

**PNRPU Mechanics Bulletin**. 2024;(3):57-64

ENERGY CHARACTERISTICS OF THE PASSAGE OF A SOUND WAVE THROUGH REGIONS OF STEPWISE CHANGES IN THE WAVEGUIDE CROSS SECTION

#### Abstract

At present time, the theoretical basis for modeling sound propagation in marine waveguides is the anal-ysis of boundary value problems for the Helmholtz equation, while the ocean bottom is an uneven interface between different media and is considered as a set of geological objects with different shapes and structures. The paper presents an analytical solution to the problem of sound propagation by a point source in a waveguide with a stepwise change in the cross section, which is modeled as a cylindrical protrusion or cavity. The velocity potential is constructed in each part of the waveguide decomposition as a series in normal modes, followed by matching the solution at the boundary. To calculate the unknown coefficients of normal modes, the theory of infinite systems of linear algebraic equations is used. The presented solution makes it possible to significantly simplify the study of the most important characteristic of the sound field is the energy flux through the cross section. The paper investigates the energy characteristics of a sound wave in a waveguide having a cylindrical protrusion (or cavity). Examples of numerical implementation are given for problem parameters, which are typ-ical for geophysical waveguides.

**PNRPU Mechanics Bulletin**. 2024;(3):65–74

THE STATE OF STRESS AND DESTRUCTION OF AN ADHESIVE WHEN JOINING PLATES WITH A LAP

#### Abstract

The deformation of an adhesive layer of finite thickness connecting two bodies with an overlap in a linear elastic formulation is considered. The stressed state of the layer is considered on the basis of the average thickness and the associated equilibrium conditions of boundary stresses. The deformed state of the layer is determined by its boundary displacements. Based on the system of variational equilibrium equations for the composite coupled by the displacement field of the adhesive layer, a numerical solution to the problem was obtained using the finite element method. To approximate the displacement field of load-bearing bodies, allowing to take into account tensile and compressive deformations in two orthogonal directions, an analytical solution to the corresponding problem is obtained. The qualitative similarity of solutions for average stresses in the layer is shown in comparison with the solution within the framework of classical plate theory. A comparison is made of the known analytical concepts for this problem, the obtained numerical and simplified analytical solutions. Taking into account the change along the length of the layer of average stress, orthogonal to the separation of the layer at a finite thickness, in the proposed formulation of the problem can affect the value of the boundary tangential stresses, and a change in the average shear stress of the layer leads to a difference in the separation stresses along the boundaries of the adhesive layer. This effect cannot be taken into account in models that use the hypothesis of homogeneity of the stress state throughout the layer thickness without taking into account boundary stresses. Using the boundary stresses of the layer introduced into the model as criterion characteristics, it is possible to simulate the detachment of the adhesive from the load-bearing bodies along the mating surfaces. It is shown that for the problem under consideration, achieving the criterion characteristics for detachment and shear leads to destruction of the adhesive layer along identical surfaces.

**PNRPU Mechanics Bulletin**. 2024;(3):75-84

NUMERICAL AND EXPERIMENTAL INVESTIGATIONS OF DYNAMIC DEFORMATION AND FAILURE ON IMPACT OF RECTANGULAR PLATES

#### Abstract

In solving of reliability and safety problems of the modern structures, especially in extreme conditions, it is very important to have reliable failure criterion. In a number of areas, such as aircraft industry, nuclear power industry, etc. regulatory documents set the requirements to keep strength of the products under high local dynamic loadings. In such local impacts on the basis of numerical calculation it is possible to select the model experiments in which a history of deformation in the most stressed point is very close to the real structure. A dynamic deformation and failure of a rectangular aluminum plate of constant thickness impacted by titanium plate in the range of V0=160.8…195.0 m/s is considered in this paper. A numerical solution of the elastic-plastic problem is conducted by finite element method with explicit scheme of integration in time. The true stress-strain curves of the materials in strain rate range of ε ̇=10^(-3)…10^4 s^(-1) are used, the friction between elements is modelled with friction coefficient f =0.1…0.2. By investigation of convergence of the numerical solutions the needed size of finite element is defined Δ=1mm. Based on numerical results, agreed with experimental deep of shear of the target, the limit level of effective plastic strain was obtained εр1= 2.5δ (δ-material elongation) in three axial compression condition. Destruction of the plate from the opposite side is realized close to two axial tension conditions reaching the limit level of effective plastic strain of εр2= 1.17δ. In frame of experimental-computational approach it is discovered that for through penetration it is necessary to reach of limit levels of effective plastic strain as from impact side of the plate as from opposite side of the plate.

**PNRPU Mechanics Bulletin**. 2024;(3):58-96

FINITE-DIFFERENCE ANALYSIS OF PLANE-PARALLEL AND PLANE-RADIAL FLOWS IN THE ELASTIC MODE OF LIQUID AND GAS FILTRATION

#### Abstract

The theory of filtration of liquids and gases through porous media has historically been used to solve a large number of applied problems, ranging from the movement of groundwater to the regularities of consolidation of biological tissues. This work examines two basic problems of the physics of oil and gas reservoirs - plane-parallel and plane-radial flows of liquid and ideal gas. Darcy's linear law is used as the equation of fluid motion. The constitutive equations for the skeleton include a term that takes into account the effect of fluid pressure on its deformation. The constitutive equations for the fluid take account of the influence of the skeleton on the compressibility of the fluid. In this way, a coupled problem is formulated for the elastic regime of fluid filtration. The model is verified based on analytical solutions, as well as numerical solutions obtained by other authors. It is shown that the obtained numerical solutions, namely, the distributions of pore pressure, filtration rates and mass flow rates, with a high accuracy coincide with analytical solutions. Additionally, fluid filtration through a quasi-isotropic medium is considered. It is shown that the presence of a layer with reduced permeability does not lead to nonlinearity in the velocity distribution for liquids and in the product of density and velocity for gas, but their values decrease. On the contrary, the pressure distribution profile changes abruptly when moving from layer to layer. Formulas are proposed for determining the effective permeability of a medium based on numerical simulation data of a plane-parallel flow. The results obtained can be used in calculating the operating parameters of oil and gas fields.

**PNRPU Mechanics Bulletin**. 2024;(3):97-107

EXPERIMENTAL VERIFICATION OF THE FREQUENCY METHOD FOR EVALUATING THE AXIAL LOAD AND IMPERFECTION OF BOUNDARY CONDITIONS IN REINFORCING BARS

#### Abstract

Evaluation of the forces and stiffness of joints is of fundamental importance for the control of rod elements of fastenings of the roof of mines, bridges, mesh shells and other structures. Existing evaluation methods are divided into static and dynamic. The paper considers a method for dynamic estimation of the longitudinal force and angular stiffness coefficients of the embedment of an imperfectly fixed rod according to the spectrum of its bending vibrations by comparing the experimentally recorded vibration frequencies with the theoretical frequency spectrum generated on the basis of the models of the Timoshenko beams. To match the results of the theoretical model with a set of experimental frequencies, a combination of well-known optimization algorithms based on global search and local minima is used. This approach is actively used, in particular, to analyze the working conditions of reinforcing bars in historical stone structures. In this work, an experimental verification of the dynamic technique was carried out on rod models with known values of the longitudinal force and the angular stiffness of the embedment. For this, two rods are considered as model samples. In one of them, predetermined longitudinal forces were created during tension in a testing machine. The other beam had the form of a cantilever with an end threaded fastening, according to the degree of tightening of which, controlled by the static deflection of the cantilever, the angular stiffness coefficient of the fastening was determined. As a result, depending on the parameters of the beam, the minimum number of recorded natural frequencies of its oscillations is determined, which is necessary for the dynamic estimation of the longitudinal force with an acceptable error. Also demonstrated is the dependence of the critical frequency of the Timoshenko beam from the internal force factor - the tensile force.

**PNRPU Mechanics Bulletin**. 2024;(3):108-117