## No 1 (2021)

**Year:**2021**Articles:**10**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/131**DOI:**https://doi.org/10.15593/perm.mech/2021.1

Effectiveness of 2-D and 3-D modelling of dovetail joint of composite fan blade for choosing rational reinforcement schemes

#### Abstract

Turbofan jet engines are among the most complex and responsible constructions in the world. The creation of modern globally competitive engines is impossible without the use of digital twin technologies: a set of computational models that fully describe the behaviour of the structure under any operating conditions. Today, composite materials are widely used in many industries. In aircraft engines, their use is very promising for fan blades and fan case to reduce the overall weight of the engine and inertial loads. The dovetail joint of the fan blade works in compound stress conditions. To assess the strength of this element, it is necessary to consider a three-dimensional formulation of the problem, which requires significant computational resources. The use of composite materials is complicated by the complexity of preparing mesh models. A correct choice of the material strength criterion is another important factor that must be taken into account during the analysis of the mechanical behaviour of the thick-walled composite structures. The chosen criterion largely determines the reliability and weight efficiency of the composite structure. This paper considers the possibility of replacing the three-dimensional statement of the problem with a two-dimensional one when choosing rational reinforcing schemes for the dovetail joint of a CFRP fan blade at the initial stages using Daniel strength criterion.

**PNRPU Mechanics Bulletin**. 2021;(1):5-11

Analysis of the stability of the computational algorithm to a change in the geometric parameters of cylindrical shell structures

#### Abstract

This study deals with testing sustainability of a computational algorithm to a change in geometric parameters of cylindrical shell structures. A change in geometry implies the replacement of one type of a cylindrical shell (elliptic, hyperbolic, parabolic) with another so that the quantitative change (the difference in elevations) in the area under consideration is minimal. On the one hand, this test allows to assessing the correctness of the algorithm itself and is relevant for algorithms that use both numerical methods and symbolic calculations. On the other hand, it allows to evaluating the possibility of simplifying calculations by approximating a complex surface with a simpler one both in understanding the surface definition itself and in expressing its basic characteristics such as Lame coefficients and main curvatures. A mathematical model of deformations of shell structures based on the hypotheses of Timoshenko (Mindlin - Reisner) are used in the work. The model takes into account transverse shifts, geometric nonlinearity and orthotropy of the material, and its written in the form of a functional of the total potential strain energy. The calculation algorithm is built on the basis of the Ritz method to reduce the variational problem of the minimum functional to the solution of a system of nonlinear algebraic equations, and on the method of continuing the solution with the best parameter for its solution. All calculations were carried out in dimensionless parameters. Three types of cylindrical panels are calculated, and critical loads of buckling and deflection fields at subcritical and supercritical moments are obtained. It is shown that for the considered class of problems the previously proposed mathematical model and computational algorithm are resistant to changes in the geometry of the structure.

**PNRPU Mechanics Bulletin**. 2021;(1):12-21

Studying Deformation Behaviors in Austenitic Stainless Steels within a Temperature Range of 143 K < T < 420 K

#### Abstract

This work deals with studying staging and macroscopic strain localization in austenitic stainless steel 12Kh18N9T within a temperature range of 143 K < T < 420 K. The visualization and evolution of macroscopic localized plastic deformation bands at different stages of work hardening were carried out by the method of the double-exposure speckle photography (DESP), which allows registering displacement fields with a high accuracy by tracing changes on the surface of the material under study and then comparing the specklograms recorded during uniaxial tension. The shape of the tensile curves σ(ε) undergoes a significant change with a decreasing temperature due to the γ-α'-phase transformation induced by plastic deformation. The processing of the deformation curves of the steel samples made it possible to distinguish the following stages of strain hardening, i.e. the stage of linear hardening and jerky flow stage. A comparative analysis of the design diagrams (with the introduction of additional parameters of the Ludwigson equation) and experimental diagrams of tension of steel 12Kh18N9T for different temperatures is carried out. The analysis of local strains distributions showed that at the stage of linear work hardening, a mobile system of plastic strain localization centers is observed. The temperature dependence of the parameters of plastic deformation localization at the stages of linear work hardening has been established. Unlike the linear hardening, the jerky flow possesses the propagation of single plastic strain fronts that occur one after another through the sample due to the γ-α' phase transition and the Portevin-Le Chatelier effect. It was found that at the jerky flow stage, which is the final stage before the destruction of the sample, the centers of deformation localization do not merge, leading to the neck formation.

**PNRPU Mechanics Bulletin**. 2021;(1):22-30

Compensatory bellows oscillations as a corrugated shell with liquid inside

#### Abstract

The paper deals with a relevant problem of shipbuilding, i.e. calculation of free and forced vibrations of pipeline compensatory bellows. These devices are used to reduce the vibration load caused by ship power machines. When analyzing the vibrations of the compensatory bellows, it is necessary to take into account the liquid contained in the bellows. In this work, the design model of the bellows is represented by a corrugated elastic shell as a material surface with five degrees of freedom. A variant of the classical theory of shells, built on the basis of Lagrangian mechanics, is used. The influence of the liquid is taken into account by two models. First, the liquid is considered to be ideal and incompressible and is considered through the attached mass to the shell. The shell is replaced by a cylindrical surface with a radius in the middle line of the corrugation. To account for the influence of the frequency of bellows oscillations on the attached inertia of the liquid in the calculation we also used the acoustic approximation; and derived a formula for a generalized attached mass of the ideal compressible liquid. The equations of the bellows oscillations under the periodic loading are obtained. The problem has been solved by the finite difference method. The values of natural frequencies of free vibrations are obtained for the compensatory bellows from the corrosion-resistant heat-resistant steel. It is shown that by taking account of the liquid, we significantly change the natural frequencies of the bellows. With high-frequency vibrations it is necessary to take into account the compressibility of the liquid. The problem of the forced vibrations of the bellows caused by a displacement of its end face by the harmonic law is solved. The internal forces and moments are determined, as well as occurring stresses by Mises criterion in the bellows. We found the critical value of the end face displacement at a frequency of 50 Hz, at which the bellows goes into a plastic state.

**PNRPU Mechanics Bulletin**. 2021;(1):31-40

Influence of preliminary thermal aging on the residual interlayer strength and staging of damage accumulation in structural carbon plastic

#### Abstract

Aging of composites is a pervasive problem that leads to mechanical properties degradation, reduced design life of a structure and premature accidental failure. The work is devoted to an experimental study of the preliminary temperature aging effect on the residual mechanical properties of structural CFRP. The joint use of test systems and systems for registration and analysis of acoustic emission signals was applied. The Short Beam Shear Test of CFRP specimens were carried out using the short beam method. The tests were carried out on universal electromechanical systems Instron 5882 and Instron 5965 in accordance with the recommendations of ASTM D2344. In the process of loading the samples were continuously recorded by using the acoustic emission signals system AMSY-6. A piezoelectric sensor with a frequency range of 300-800 kHz was used. The test and diagnostic systems were synchronized during the tests. In the course of the work the values of the interlayer shear strength were determined for the samples of CFRP. Typical types of the sample destruction are illustrated. When analyzing the change in the mechanical properties of the carbon fiber reinforced plastic from a temperature increase the critical values of temperatures were established in which a sharp decline in the strength and elastic characteristics of materials occurs due to an active destruction of the binder. The graphs of the energy parameter dependence and frequency characteristics of acoustic emission signals on time have been constructed and analyzed. The estimate of the processes of damage accumulation in composites is carried out. The change of the damage accumulation mechanisms was illustrated. The obtained results illustrate the effect of elevated temperatures and the duration of their impact on the mechanical behavior of structural CFRP specimens during the static tests for the interlayer shear.

**PNRPU Mechanics Bulletin**. 2021;(1):41-51

Constitutive relations for materials with strain state dependent properties

#### Abstract

Many materials demonstrate a dependence of mechanical properties on the type of stressed or deformed states. This is most noticeable in the dependence of the processes of shear and bulk deformation. Such materials include rocks, structural graphite, concrete, some grades of steel, cast iron, and aluminum. The main properties of these materials are an absence of a "single curve" relationship between the intensity of stresses and the intensity of deformations. Under shear conditions, bulk deformations can occur. Such materials can be described by constitutive equations that depend on the parameter of the type of a stress state, which is the ratio of the first invariant of the stress tensor to the stress intensity. Thus, these defining relations give the dependence of the strain tensor components on the stress tensor components. Such defining relations can be quite cumbersome, and therefore do not allow an analytical treatment to obtain defining relations that give the dependence of the components of the stress tensor on the components of the strain tensor. The paper proposes the constitutive relations obtained from the analysis of test results of various materials, which properties depend on the type of deformed state. Conditions are derived for material constants that ensure the uniqueness of the solution of boundary value problems. Based on experimental data obtained under the conditions of the proportional loading of various rocks: limestone and talcochlorite, as well as the results of mechanical tests of several grades of concrete, the constants of the mathematical model are determined. The results of the experimental studies are compared with theoretical dependencies predicted by the model. The limited applicability of the proposed constitutive relations is established.

**PNRPU Mechanics Bulletin**. 2021;(1):52-62

Contact problems for an inhomogeneous elastic wedge with variable Poisson’s ratio

#### Abstract

Plane contact problems of the elasticity theory are investigated for a wedge when Poisson’s ratio is an arbitrary smooth function with respect to the angular coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the angular coordinate. A finite contact domain is given on one wedge face, it does not include the wedge apex, while the other wedge face is rigidly fixed (problem A) or stress-free (problem B). To reduce the problems to integral equations with respect to the contact pressure, we use the general Freiberger type representation for the solution of elastic equilibrium equations written in polar coordinates with variable Poisson’s ratio. Exact solutions of auxiliary problems are constructed with the help of Mellin integral transforms. The regular asymptotic method employed is effective for contact domains relatively distant from the wedge apex. It is shown that logarithmic terms appear in the asymptotic solutions for the inhomogeneous material which are missing in the well-known asymptotics for the homogeneous one. In contact problem C which is corresponding to problem A, the friction and roughness are taken into account in the contact region. The roughness of the wedge surface is simulated by a Winkler type coating. The collocation method is used for solving integral equations of the second kind. Unlike problem A, in problem C the contact pressure does not have square root singularities at end-points where it takes finite values. Calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the wedge.

**PNRPU Mechanics Bulletin**. 2021;(1):63-71

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II

#### Abstract

In this study coefficients of the multi-parameter Williams power series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending are obtained by the use of the finite element analysis. In SIMULIA Abaqus, the finite element analysis software, the numerical solutions for these two cracked geometries are found. The rectangular plate with the central crack has the geometry similar to the geometry used in the digital photoelasticity. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The numerical solutions obtained are utilized for the determination of the coefficients of the Williams series expansion. The higher-order coefficients are extracted from the finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. Determination of the coefficients of the terms of this series is performed using the least squares-based regression technique known as the over-deterministic method, for which stresses data obtained numerically in SIMULIA Abaqus software are taken as inputs. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from the finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy. The higher-order terms in the Williams series expansion for the semi-circular notch disk are found.

**PNRPU Mechanics Bulletin**. 2021;(1):72-85

Interpretation of the generalized parameter of the probability of failure through the plastic stress intensity factor

#### Abstract

The main purpose of this work is to statistically analyze the fracture toughness of compact specimens made of S55C steel in terms of elastic and plastic stress intensity factors. The fracture toughness tests results at three-point bending were used for a comparative statistical analysis of the fracture parameters. Five type of specimen configuration with various thicknesses were tested at a constant ratio between crack length and specimen width. The critical loads were obtained as a tests result for various combinations of crack length and specimen thickness. In addition, uniaxial tensile tests were carried out to determine the main mechanical properties of the material. Obtained material properties were used in numerical calculations. Numerical calculations were carried out to determine the elastic and plastic stress intensity factors. Three-dimensional finite element analysis was performed on the basis of the experimental data on curvilinear crack front positions in tested specimens. The crack tip stress-strain fields were obtained for each of the tested samples as a result of numerical calculations. These fields were used to calculate the values of the plastic intensity factors along the curvilinear crack fronts. A statistical analysis of the fracture toughness of compact specimens made of S55C steel was carried out based on the obtained critical values of elastic and plastic stress intensity factors. The advantages of using the plastic stress intensity factor as a generalized parameter for the fracture probability are demonstrated. In addition, the sensitivity of the plastic stress intensity factor to constraint effects avoids the introduction of additional parameters into the statistical models.

**PNRPU Mechanics Bulletin**. 2021;(1):86-94

The analysis of flutter characteristics based on generalized parameters of eigen modes of vibrations

#### Abstract

Flutter numerical analysis of a dynamically scaled model (DSM) of a high aspect ratio wing was performed using experimentally obtained generalized parameters of eigen modes of vibrations. The DSM is made of polymer composite materials and is designed for aeroelastic studies in a high-speed wind tunnel. As a result of the analysis, safe operation conditions (flutter limits) of the DSM were determined. The input data to develop the flutter mathematical model are DSM modal test results, i.e. eigen frequencies, mode shapes, modal damping coefficients, and generalized masses obtained from the experiment. The known methods to determine generalized masses have experimental errors. In this work some of the most practical methods to get generalized masses are used: mechanical loading, quadrature component addition and the complex power method. Errors of the above methods were analyzed, and the most reliable methods were selected for flutter analysis. Comparison was made between the flutter analysis using generalized parameters and a pure theoretical one based on developing the mathematical model from the DSM design specifications. According to the design specifications, the mathematical model utilizes the beam-like schematization of the wing. The analysis was performed for Mach numbers from 0.2 to 0.8 and relative air densities of 0.5, 1, 1.5. Comparison of the two methods showed the difference in critical flutter dynamic pressure no more than 6%, which indicates good prospects of the flutter analysis based on generalized parameters of eigen modes.

**PNRPU Mechanics Bulletin**. 2021;(1):95-102