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7UDC: 539.3
STATE-OF-THE-ART IN THE INVESTIGATION OF THREE-LAYER RODS
Anarova Shahzoda Amanboyevna TATU, t.f.d. dotsent, [email protected], 90 322 66 71
Shokirov Davron Abdug'affor o'g'li NamMQI, doktorant, [email protected], 94 613 60 59
Amonov Otabek Tulkunovich "Atom elektrostantsiyani qurish Direktsiyasi" davlat unitar korxonasi direktori 712020920
amonovotabek@mail. ru
Annotatsiya. Ushbu maqola sharh xarakteriga ega; tadqiq etilayotgan muammo yuzasidan mamlakatimiz va xorijiy mamlakatlar olimlari tomonidan olib borilgan tadqiqotlar tahlil qilinadi. Maqolada uch qatlamli sterjenlarni o'rganishning hozirgi holati muhokama qilinadi. Ushbu masala bo'yicha butun dunyoda olib borilgan ilmiy tadqiqotlar batafsil o'rganilgan va maqolada keltirilgan. Adabiyot manbalarini tahlil qilish asosida uch qatlamli sterjenlarning deformatsiya holatini o'rganish dolzarb muammolardan biri ekanligi aniqlandi. Hozirgi vaqtda uch qatlamli sterjenlarning kuchlanish-deformatsiya holatini va barqarorligining kritik parametrlarini hisoblash masalalariga katta e'tibor berilmoqda. Biroq, ushbu masala bo'yicha nashr etilgan maqolalarda, amaliyot uchun muhim bo'lgan uch qatlamli tuzilmalarning ba'zi yuklash sxemalari hisobga olinmaydi. Ma'lumki, uch qatlamli konstruktiv elementlar zamonaviy sanoat tarmoqlarida keng qo'llaniladi, shuning uchun ularni yangi ish sharoitlarida hisoblash usullarini ishlab chiqishga doimiy ehtiyoj bor. Uch qatlamli sterjenning kinetik va potentsial energiyasining o'zgarishi va tashqi jism hamda sirt kuchlari ishining o'zgarishi aniqlanadi. Kinetik va potentsial energiyaning o'zgarishini va tashqi jism hamda sirt kuchlarining ishini aniqlash uchun Ostrogradskiy-Gamilton printsipi qo'llaniladi.
Аннотация. Данная статья носит обзорный характер; анализируются исследования, проведенные учеными нашей страны и зарубежных стран в области изучаемой проблемы. В статье рассматривается современное состояние исследования трехслойных стержней. Подробно изучены и представлены в статье научные исследования, проводимые по данному вопросу во всем мире. На основе анализа литературных источников определено, что исследование напряженного состояния трехслойных стержней является одной из актуальных проблем. В настоящее время большое внимание уделяется вопросам расчета напряженно-деформированного состояния и критических параметров устойчивости трехслойных стержней. Однако в статьях, опубликованных по данному вопросу, не рассматриваются некоторые важные для практики схемы нагружения трехслойных конструкций. Известно, что трехслойные конструктивные элементы широко используются в современных производствах, поэтому существует постоянная потребность в разработке методов их расчета в новых условиях эксплуатации. Определены вариации кинетической и потенциальной энергии и вариации работы внешнего тела и поверхностных сил трехслойного стержня. Принцип Остроградского-Гамильтона применяется для определения изменения кинетической и потенциальной энергии и работы внешних объемных и поверхностных сил.
Abstract. This article is of a review nature; the research conducted by the scientists of our country and foreign countries in the field of the problem under study is analyzed. The article discusses the current state of the study of three-layer rods. Scientific research conducted on this issue worldwide is studied in detail and presented in the article. Based on the analysis of literature sources, it was determined that the study of the strain state of three-layer rods is one of the urgent problems. At present, much attention is paid to the issues of calculating the stressstrain state and critical parameters of the stability of three-layer rods. However, in the articles published on this issue, some loading schemes of three-layer structures that are important for practice are not considered. It is known that three-layer structural elements are widely used in modern industries, so there is a constant need to develop methods for their calculation in new operating conditions. Variations of kinetic and potential energy and variations of the work of external body and surface forces of a three-layer rod are determined. The Ostrogradsky-Hamilton principle is applied to determine the variation of kinetic and potential energy and the work of external body and surface forces.
Kalit so'zlar: uch qatlamli sterjen, Ostrogradsky-Gamilton printsipi, kinetik energiya, potensial energiya, deformatsiya, ko'chish, kuchlanish, hajmiy kuchlar, tashqi kuchlar, chetki kuchlar.
Ключевые слова: трехслойный стержень, прицип Остроградского-Гамильтона, кинетическая энергия, потенциальная энергия, слой, деформация, перемещения, напряжений, объемных сил, поверхностные силы, торцовые силы.
Keywords: three-layer rod, Ostrogradsky-Hamilton principle, kinetic energy, potential energy, layer, deformation, displacements, stresses, body forces, surface forces, end forces.
1. Introduction
Recently, the demand for the use of layered thin-walled structural elements in aircraft, rocket-, machine-, instrumentation and ship-building, in energy carriers production and transportation has significantly increased. This necessitates the development of mathematical models and methods for their calculation under various types and forms of loads.
Methods for calculating three-layer plates and shells are considered in detail in [1]. There are numerous publications in which the theory of multilayer plates and shells is built on the basis of hypotheses regarding the strain pattern of the multilayer structure (a package) as a whole. Based on a unified approach, the book presents the theory of multilayer structures with an arbitrary number of layers and with relatively broad assumptions about the properties of an individual layer. The theory is applicable not only to thin and thin-walled structures but also to bodies of arbitrary shape. The methods for calculating systems of a regular structure, which in some cases make it possible to construct solutions containing the number of layers as a parameter, are developed in the book, and the theory of layered composite materials is obtained as a result of passing to the limit to systems with a fine layered structure.
At present, much attention is paid to the issues of calculating the stress-strain state and critical parameters of the stability of three-layer plates and shells. However, the articles published on this topic do not consider some practical loading schemes for three-layer structures, for example, loading with pressure non-uniformly distributed over the surface of a cylindrical shell, a combination of loads with an axial force, and one-layer loading. There are no calculated dependences for determining the stress-strain state of orthotropic shells in the zones of action of edge effects; the strength and stability of three-layer plates and shells reinforced with a set. The study in [2] summarizes the results of well-known publications and new methods that allow calculating the parameters of the stress-strain state and critical loads of three-layer plates and cylindrical shells. The results of some previously considered problems, the solution of which
does not agree with the experiment, are refined. The calculation dependencies for plates and shells with orthotropic bearing layers and a core were developed; they make it possible to calculate three-layer structures made from new structural materials. Most of the proposed methods were experimentally tested with a large number of samples.
A rigorous kinematic analysis gives a general idea of the change in displacement across the thickness of multilayer plates, which allows a discontinuous distribution of displacement across each interface of adjacent layers to allow the inclusion of surface imperfection effects [3]. The spring-layer model, which has recently been effectively used in the field of micromechanics of composites, has been introduced to model imperfectly coupled interfaces of multilayer plates. The linear theory underlying the dynamic response of non-uniformly weakened multilayer anisotropic plates is presented on the basis of the Hamilton principle. This theory has the same advantages as conventional higher-order theories over classical and first-order theories. In addition, when modeling interfacial properties, the conditions for imposing thrust continuity and a displacement discontinuity along each interface are used. In the particular case of vanishing interface parameters, this theory is reduced to the recently well-developed zigzag theory. A closed-form solution is presented as an example, and some numerical results are plotted to illustrate the effect of interface weakness.
In [4], a general method for determining the effective elastic properties of two-dimensional cellular multilayer cores with an arbitrary topology and cell geometry is presented. The scheme uses the strain energy-based representative volume element procedure, assuming that macroscopically equivalent strain states should cause the same strain energy in the representative volume element, regardless of whether a real microstructure or an "effective" homogenized medium is considered. The strain energy can be estimated either analytically or numerically. Both approaches agree well in a number of examples considering different geometries of the sandwich core.
In [5], axisymmetric resonant vibrations of an elastic round three-layer plate under the action of local periodic surface loads of rectangular, sinusoidal, and parabolic shapes were studied. The hypotheses of broken normal were used to describe the kinematics of a plate not symmetrical in thickness. The core was assumed light. The initial-boundary value problems were solved analytically. Solutions were analyzed.
In [6], forced vibrations of an elastic three-layer beam were considered under the action of local loads of rectangular and parabolic shapes. It was assumed that the hypothesis of broken normal describes the kinematics of a package of asymmetric thickness. The core was assumed hard and compressible. Analytical solutions to problems were found under pulsed and resonant impacts. A numerical analysis was performed. The results were compared with the case of a local shallow load of rectangular form.
Statements and methods for solving problems of statics for a wide class of metal-polymer systems under complex force, thermal, and radiation effects are presented in [7]. The rheonomic and plastic properties of the layer materials were taken into account. A number of analytical and thermal solutions for three-layer metal-polymer rods, plates, and shells are presented.
Recently, three-layer structures, consisting of two bearing face layers and a core that ensures their joint work, have become widely used. Under bending strain, three-layer structures turn out to be the most rational, that is, close to optimal in terms of ensuring a minimum of weight indices for given restrictions on strength and rigidity.
The theory of multilayer structures can be interpreted as a result of the generalization of the classical theory of plates and shells in the theory of three-layer structures. In a number of cases, multilayer structural elements can no longer be considered thin in the sense of the
hypotheses of the classical theory. With an increase in the number of layers and the use of various cores, the effects associated with the operation of individual layers begin to play a significant role. In addition to transverse shears and compression of normals, it is often necessary in multilayer structures to take into account moment effects in load-bearing layers, local forms of buckling, etc.
Rods, plates, and shells of a layered structure are usually combined from materials with significantly different physical and mechanical properties. Bearing layers made of materials of high strength and rigidity are designed to absorb the main part of mechanical load. Binder layers, which serve to form a monolithic structure, ensure the force redistribution between the bearing layers. Another group of layers is designed to protect against thermal, chemical, radiation, and other undesirable influences. This combination of layers makes it possible to ensure reliable operation of systems in adverse environmental conditions, and to create structures that combine high strength and rigidity with a relatively low weight [8].
In [9], the bending of a round multilayer plate with a light core, resting on an elastic foundation, was considered. To describe the kinematics of a package of asymmetric thickness, the hypothesis of broken normal was adopted. The response of the foundation was described based on the Winkler model. Loading was local and symmetrical. Combined equilibrium equations and analytical solutions for displacements were obtained. Numerical results for a multilayer metal-polymer plate were given.
The bending of an elastic rectangular three-layer plate with a rigid core resting on an elastic foundation was considered in [10]. To describe the kinematics of the bearing layers, the Kirchhoff hypotheses were adopted. The response of the foundation was described by the Winkler model. A system of equilibrium equations and its solution in displacements were obtained. Numerical results were given for a rectangular three-layer plate.
In [11], thermoelastic bending of an annular multilayer plate with a light core, lying on an elastic foundation, was considered. To describe the kinematics of a package of an asymmetric thickness, the hypothesis of broken normality was adopted. The response of the foundation was described on the basis of the Winkler model. Thermal force loading was local and symmetrical. Combined equilibrium equations and analytical solutions in displacements were obtained. Numerical results for a multilayer metal-polymer plate were given.
The strength of materials is one of the sections of the mechanics of a deformable rigid body [12]. This is an extremely important engineering discipline, necessary for engineers of any specialty. Without fundamental knowledge in this area, it is impossible to create various kinds of machines and mechanisms, to erect industrial and civil structures, bridges, power lines, antennas, hangars, ships, aircraft, units of nuclear power plants, rocket and jet technology, etc.
In [13], the thermomechanical bending of an elastic-plastic annular (solid or ring) multilayer plate made with a light core, resting on an elastic foundation, was considered. The hypotheses of broken normal were used to describe the kinematics of a package of plates not symmetrical in thickness. The main response was described by the Winkler model. A system of equilibrium equations and their exact solutions in displacements were obtained. Numerical results for a multilayer annular metal-polymer plate were also presented.
A closed system of constitutive equations for dynamic and geometric quantities in a fluid-saturated viscoelastic porous medium was constructed in the framework of the three-dimensional theory of elasticity given in [14]. For the first time, using the mathematical theory of discontinuities, expressions were obtained for intensity waves in porous media.
The main aim of the research presented in [15] was to study the behavior of an elastic line under loading of a three-layer beam based on analytical formulas constructed using the theory of
bending of a multilayer beam, described by V.V. Vasiliev, the theory of loading of three-layer shells by E.I. Grigolyuk, and the method of nonlinear finite elements (FEM) under the bending of an elastic rod in the framework of the nonlinear dynamics of flexible rods, described by F.N. Shklyarchuk and T.V. Grishanina. The appearance of non-trivial types of elastic lines under the loading of a beam with varying rigidity of the layer was identified and explained.
In [16], the thermal force bending of an elastic-plastic three-layer beam with a hard core was studied; the beam was connected to an elastic foundation. To describe the kinematics of a package, asymmetric in thickness, the hypothesis of broken normal was adopted. The Winkler model described the response of the foundation. A system of equilibrium equations and their exact solutions in displacements were obtained, and numerical results for a three-layer metal-polymer beam were presented.
In [17], the formulations and methods for solving problems of statics and dynamics of three-layer structural elements associated with an elastic foundation under complex force, thermal, and radiation effects were systematically presented. The physically nonlinear properties of the material of the layers were taken into account. A number of analytical solutions and their numerical parametric analysis for three-layer rods and plates were presented.
The authors of the textbook [18] based their studies on the courses "Theory of Elasticity" and "Theory of Elasticity and Plasticity", which they taught to the students of engineering specialties for a number of years at the Moscow State Aviation Institute (at present, Moscow State Aviation Technology University) and the Belarusian State University of Transport. The main purpose of the book is to present modern approaches to the formulation and solution of problems of theories of elasticity, plasticity, and viscoelasticity. The textbook is intended, first of all, for students of technical universities, therefore, along with the tensor form of notation, the coordinate form was used. The authors were based on a deep conviction that such material for any specialties should be presented with a sufficient level of mathematical rigor and evidence, where it is required within the course. In this regard, the book can be used in the educational process for a wide range of specialties. For the successful mastering of the material, the knowledge of the relevant sections of the standard course of mathematical analysis, linear algebra, and differential equations is necessary.
Symmetric transverse vibrations of an elastic round multilayer plate under the action of thermal and ionizing radiation were studied in [19]. The plate stands on a inertia-free Winkler base. The face layers are described by the Kirchhoff hypotheses, and the light core is described by the hypothesis of broken normal. Analytical solutions were obtained. The numerical results were analyzed.
In [19], models of elastic, elastic-plastic, viscoelastic-plastic, and thermoviscoelastic-plastic media are considered. Fundamental solutions of the theory of elasticity, dynamic problems, problems of geomechanics, methods for obtaining the thermomechanical characteristics of materials, and methods and examples of solving problems, including nonlinear bending and vibrations of three-layer plates, are presented.
Symmetric transverse vibrations of a round multilayer metal-polymer plate under the action of thermal shock were studied in [21]. The plate was connected to the inertia-free Winkler foundation. It was assumed that the face plates satisfy the Kirchhoff hypothesis, and the strained normal in the low-density core is rectilinear and incompressible in thickness. Analytical solutions were obtained and their numerical analysis was given.
Non-homogeneous structures are widely used in various fields of mechanical engineering and construction; therefore, it is important to develop methods for their strength calculation under various loads. The study in [22] presents the results on a single quasi-static and dynamic
strain of three-layer structural elements connected/not connected to a Winkler foundation. Here the bending with the tension of a three-layer rod asymmetrical in thickness in a temperature field was considered.
In [23], the strain of a physically nonlinear three-layer rod under cyclic loading in a temperature field was considered. To describe the kinematics of a rod package, asymmetric in thickness, the hypotheses of broken normal were adopted. A technique for solving the corresponding boundary value problems was proposed. Analytical solutions were obtained for the problems of thermo-elasticity and thermo-elastic-plasticity under direct and reverse loading. A numerical analysis of solutions was conducted.
In [24], a finite element formulation of geometrically exact multilayer beams was proposed. Interlayer slip and uplift were not taken into account. The number of layers was arbitrary, and the horizontal and vertical displacements of the base axis of the composite beam and the rotation of each layer in the cross-section were taken as the basic unknown functions. Due to the geometrically exact formulation of the problem, the constitutive equations are nonlinear with respect to the main unknown functions, and the solution is obtained numerically. In general, each layer can have different geometric and material properties but since the layers are rigidly connected, this model is applied mainly to homogeneous layered beams. Numerical examples compare the results of the present model with existing geometrically nonlinear models of sandwich beams, with two-dimensional elements of plane stress, and, where applicable, with the results of the theory of elasticity. Comparison with 2D plane stress elements shows that the layered beam model is very effective for modeling thick beams where cross-sectional strain needs to be taken into account.
Free vibrations of a three-layer cylindrical circular shell in an elastic medium were considered in [25]. It was assumed that the bearing face layers satisfy the Kirchhoff-Love hypotheses. In a thick core, the work of transverse shear and compression in thickness was taken into account. It was assumed that the displacement variations are linear in the transverse coordinate. At the contact boundaries, the conditions for the continuity of displacements were used. The Winkler hypothesis was taken for an elastic inertia-free medium. The change in eigenfrequencies of oscillations depending on the rigidity characteristics of the shellenvironment system was studied.
The effect of the temperature field on the strain of a three-layer elastic rod with a compressible core was considered in [26]. To describe the kinematics of a package asymmetric in thickness, the hypotheses of broken normal were adopted: in thin bearing layers, Bernoulli's hypotheses are valid; in a core compressible in thickness, Timoshenko's hypothesis is fulfilled. The work of the core in the tangential direction was taken into account. The equilibrium equations were obtained using the variational method. A technique for solving the corresponding boundary value problems was proposed. Analytical solutions in displacements were obtained and their numerical analysis was conducted.
The study in [27] considers the strain of a three-layer metal-polymer rod with elastic-plastic bearing layers and a physically nonlinear core under the impact of local rectangular and sinusoidal loads. To describe the kinematics of a rod package asymmetric in thickness, the hypotheses of broken normal were adopted. The formulation of boundary value problems and methods for their solution are given. Analytical solutions were obtained in iterations of problems in the theory of small elastoplastic strains. A numerical analysis of solutions was conducted.
Currently, software systems based on the finite element method (FEM) are widely popular among engineers and scientists. In [28], the main concept of the FEM is the direct discretization of the calculated system, partitioned by the computational grid into finite elements.
The article deals with the convergence of the FEM for plate elements. Based on the assumptions made, a numerical experiment was performed to calculate the stress-strain state of a three-layer beam and a comparison of the results with theoretical data obtained from the results of the calculation of three-layer beams (TLB) was made according to the method described in the reference book by V.N. Kobelev.
In [29], the formulations and methods for solving the problems of quasi-statics of three-layer rods of the rectangular cross-section with smooth and stepped surfaces are presented. The physically nonlinear properties of the materials of face layers were taken into account under complex force, thermal, and radiation effects. A number of analytical solutions and a numerical parametric analysis of the stress-strain state of the rods under study were presented.
At present, multilayer (including three-layer) structural elements are widely used in engineering and construction. Rods, plates, and shells of a layered structure, with relatively low weight, are able to provide the desired strength and rigidity and withstand a number of other physical effects. In this regard, the creation of calculation models of three-layer rods, using various kinematic hypotheses and complex thermal-force local loads, becomes an urgent task.
In [30], the statement and the analytical solution to the boundary value problem of thermal force loading of a three-layer rod of rectangular cross-section with a compressible core under local uniformly distributed, sinusoidal, and parabolic loads are given. Numerical approbation of the solution was performed for the case of a metal-polymer rod.
In connection with the widespread use of three-layer structural elements in construction and mechanical engineering, it becomes necessary to create adequate mechanical and mathematical models for calculating their stress-strain state. In [31], strain in a temperature field of a three-layer elastic-plastic beam with a compressible core was considered. To describe the kinematics of a package asymmetric in thickness, the hypotheses of broken line were adopted: in thin bearing layers, Bernoulli's hypotheses are valid; in a core, compressible in thickness, the Timoshenko hypothesis is fulfilled with a linear approximation of displacements along the thickness of the layer. Physical relationships between stresses and strains correspond to the theory of small elastoplastic strains. The temperature change was calculated using the formula obtained by averaging the thermophysical properties of the materials of the layers over the thickness of the rod. The system of differential equilibrium equations was obtained by the variational method. An analytical solution was obtained by the method of elastic solutions for the case of a uniform distribution of a continuous load; its numerical analysis was performed.
In [32], a method for calculating three-layer bending reinforced concrete elements was considered, taking into account the diagrams of the total strain of various layers of concrete. The formulas and calculations for the cases of the presence or absence of cracks in the tension zone were obtained.
The study in [33] presents the formulations and methods for solving boundary value problems for determining the stress-strain state of three-layer rod structural elements under single and quasi-static variable loads in thermoradiation fields. Physically nonlinear properties of the material of layers are taken into account under complex force, thermal, and radiation effects. A number of analytical solutions and a numerical parametric analysis of the stress-strain state of three-layer rods are presented.
In [34], the stress-strain state of a three-layer rod under central compression was considered. The results of analytical and numerical calculations (FEM) are compared with experimental data. It is assumed that the interaction of the layers is realized through the contact layer. The contact layer is considered a transversely anisotropic elastic medium with parameters that can be represented as a set of short elastic rods, not connected to each other and oriented
normally to the contact surface. This assumption makes it possible to obtain an analytical solution to the problem in a closed form and to avoid infinite shear stresses at the interface between layers near the end face of the model. The calculation results obtained, qualitatively and quantitatively agree with the experimental results.
The reference [35] considers the problems of calculating a three-layer rod under instantaneously increasing loads. The differential equations of oscillation of a three-layer rod under the action of dynamic loads were solved. The maximum dynamic coefficient of deflection was determined for various types of fastening of the ends of the beam, and for the transition to two-layer homogeneous rods. Calculations were conducted for various values of elastic characteristics of the bearing layers of the rod. The dependence of a dimensionless parameter on the time of load action and the reduced frequency of the rod with the dynamic coefficient of deflection was analyzed. It was determined that the stress changes in the bearing layers (depending on the dimensionless parameter on the time of load action and the reduced frequency of the rod), are similar to the pattern of change in the dynamic coefficient.
The study in [36] presents the results of testing the elements of three-layer structures with mesh lightweight cores made by 3D printing using the technology of layer-by-layer laser synthesis from polyamide. The structure of cores corresponding to the so-called pantographic mechanical metamaterials is considered, in which two systems of parallel rods are separated by a small distance and connected by transverse pins in the intersection zones. It is known that in order to describe the equivalent mechanical characteristics of such materials, it is necessary to use non-classical models of the theory of elasticity, which take into account the non-local nature of strains in the structure material under load. In this paper, three options of transverse connections in the structure of a metamaterial are considered, in which transverse pins provide transmission of forces and moments (rigid connections), of forces only (hinged connections), or no connections. Such options of cores are compared with a conventional mesh core, in which the crossed rods form a rigidly connected system such as a flat frame. The manufactured samples were tested for impact resistance according to the two-support impact bending scheme using a pendulum impact tester. It was determined that with the same dimensions of the cross-sections of the rods in the core, the samples with pantographic cores with rigid transverse joints have the highest bearing capacity under the impact. However, samples with hinged joints demonstrate unusual fracture patterns, in which the damage zone is the largest, and the development of damage occurs with the formation of many small fragments that prevent the striker from passing through the structure and increase its specific energy absorption; this makes such options of cores potentially promising for creating shock-absorbing structures.
In [37], it is shown that aluminum-glass-reinforced plastics (as a promising aviation material) possess increased characteristics of specific rigidity and strength, fatigue strength, impact resistance, and residual strength after impact. At present, GLARE aluminum-glass-reinforced plastics are used for the manufacture of fuselage skinning elements for Airbus A380 long-haul passenger aircraft, and individual elements of airframes. This study is devoted to determining the dynamic behavior of SIAL specimens operating as part of three-layer structures with foamed core. The results of experimental studies of eigenfrequencies and damping coefficients of three-layer beams made with bearing face layers of five-layer aluminum-glass-reinforced plastic SIAL and with foamed polyimide core are presented. The tests were conducted using the method of free damped bending vibrations of cantilever-fixed specimens. The dynamic parameters of three-layer beams were calculated based on the analysis of the amplitude-frequency characteristics obtained by the fast Fourier transform method. The mechanical characteristics of SIAL samples and core were preliminarily determined in static and dynamic
tests. The damping coefficient of the core was determined by the method of dynamic mechanical analysis. The shear modulus of the core was determined from the results of measuring the flexural rigidity of the fabricated three-layer beams in quasi-static three-point bending tests. On the basis of a comparison of calculated data and experimental results, it was shown that in dynamic tests there is a decrease in the flexural rigidity of three-layer samples, compared with the calculated values, which may be due to the features of changing the characteristics of the porous core under dynamic loading. The value of the damping coefficient for SIAL samples was ~0.02, for foamed core ~0.08, and for three-layer beams ~0.067 in the vibration frequency range up to 60 Hz.
In [38], the objects of study were three-layer reinforced concrete enclosing structures. The structures consist of heavy concrete B25 in the outer face layers and polystyrene concrete B1 in the core. The stress-strain state of prefabricated reinforced concrete structures was studied during cracking, taking into account the effect of contact layers between concretes of different grades. In the experimental study of multilayer concrete blocks, stereoscopic microscopy and scanning electron microscopy were used. Samples were made with time intervals ranging from 30 minutes to two hours between the placement of the previous and next concrete layer. The results of the experiments showed that with the successive layer-by-layer packing of various concretes, a contact layer was formed with the mutual penetration of aggregates into adjacent layers of concrete. The thickness of the contact layer was up to 1 cm. The contact layer affects the bonding strength of concrete layers and the stress-strain state of the structure. A model and method for calculating crack formation in three-layer reinforced concrete structures with contact interlayers based on analytical and numerical calculations were proposed. Experimental data confirmed the adequacy of the proposed calculation method. The results of calculations of three-layer reinforced concrete beams show that: (i) the difference in moments during the crack formation in the schemes of three-layer reinforced concrete beams with and without considering the contact layer can reach 9.9%; (ii) the moment of crack formation obtained by the proposed method is from 7.4% to 9.1% greater than that obtained by the section conversion scheme.
2. Formulation of the problem
In the general theory of vibrations of elastic bodies, displacement u2k) and are
the functions of four variable coordinates xi, x2, x3 and time ^
The equations of the theory of rod vibrations are derived on the basis of the Ostrogradsky-Hamilton principle. The Cauchy equations and boundary conditions were derived from the Ostrogradsky-Hamilton principle.
In general form, the Ostrogradsky-Hamilton variational principle [39,40,41,42] is written
as:
S\(K-n + A) = 0, (1)
where K, n are the kinetic and potential energies, A is the work of external body and surface forces.
Displacements of points of a three-layer rod [8, 33] are:
u
(1)
u
c —1 2 у
dw(1)
dx
w
(1)
w (c < z < c + h);
u J(2) = u2
V
z + c + ■
V
r
uf3) =
(3)
z
1+
V c у
1
1
—u, + 2 1 2
dx
\ -w{X) dx
w
(2)
= W2( c - h2 <z <-c);
+
i -
c
u2 -
\ dw(2) 4 dx
w ' = —
2
V
1 + z
c
w +
fi - -
V
c
w
2
(-c < z < c).
У
3. Main part
3.1 Determination of the kinetic energy variation in a three-layer rod
When calculating the variation of kinetic energy, we use the following relation:
\SKdt = f f dvdt+ f f^y ^L-8ufdvdt+ f f^y ^-du^dvdt.
j j j Qt T~f d t JJ-fr'
t t v k=l
Integrating by parts
fSKdt= L^ 2)
k=1
du( 2)
du ,(3)
Su®
k=1
д t
St
St
St
St
du(2)
Suf +p~u^~ Su(2) + p-^- Su(- + p-±- Suf Suf} + p
du?
S2u (3)
Su ® + p-u- Suf3 +p
-u
St
su
St
3 Su(3)
dv
—Su3(1) +p-u-Su3(2) +p
S2u(3)
3- Su-
dvdt;
(2)
(3)
(4)
rrf (d ^
-JJK at2 1 St2 dt2 ~u1 dt2 J 'St2 J ' dt2
tv V y
where p is the specific mass density of the body material (assumed constant).
Substituting expressions ux(1), ux(2), ux(3), w(1), w(2), w(3) from (2) into the kinetic
energy variations (4) we open the brackets under the variation sign after performing the integration over the sections of the rod and perform the corresponding analytical calculations: variation, differentiation, integration, and reduction of similar terms on each layer of the rod. As a result, with the introduced notation, we obtain variations in the kinetic energy for each layer separately.
3.2 Determination of the potential energy variation in a three-layer rod For the stress and strain components in the x1, x2, x3 coordinate system, we introduce the notation of stress tensors a(f), a^\ a^\ afk,), a^), a^ ) and strain tensors
s\\ * s22 > sS^, s^, ¿>23), sf- . Then for the variation of the potential energy, we have: The components of the strain tensor of a three-layer rod are [39]:
£1 =
(k)
dx
du9 (k) du,' £12 + —
( )
dx
dy
£13 =
-u(k)
dx
+
(k)
dz
Based on the G.Yu. Dzhanelidze assumptions, stress components a22(),z7i{K) and a are not zero. However, in view of their smallness in comparison with stress components
(k)
(5)
(k)
33
(k) 11 ,
a223), a1(3) ,they are ignored. Therefore, a® = 0, a<33) = 0, a%) = 0 .
<
3 3
jn=zzjj =jjvwww+w+
i=1 j=1
v t v t
(2>Ssn |2) + (3)&12 (3) + ct13 0,&13 l1) + (2)&13<2) + 0)Ssn (3) )dvdt =
= J/*,™*,™ + +Jj'<T11(3)5^11(3) + J J^1'^™ + H^2^2' +
vt vt v t v t v t
v t v t v t v t
To determine the variation of the potential energy in a three-layer rod we perform the appropriate analytical calculations: variation, differentiation, integration, and reduction of similar terms on each layer of the rod. As a result, with the introduced notation, we determine the variations of the potential energy over the layers of the rod.
3.3 Determination of the variation of the work of external and surface forces in a three-layer rod We calculate the variations in the work of external forces:
3 3 3
jSAdt = p(k) 8ulk)dV + q(k) 8ulk)ds + f(k) 8ulk) ds,, (6)
V i=1 X s.
) are the components of body forces, referred to as a unit of volume, q(k) are the surface
-fi k )
forces, referred to as a unit area of the rod surface, J i are the end face forces [1], [4], [11].
Displaceme external forces (6):
l k )
Displacement expressions Ui from (2) are introduced into the variation of the work of
J 5Adt = J J £ P<(k)5u\k'dvdt + J J X P(k'«u2 'dvdt + J J £ P3(k'«u<k'dvdt
t t v k=1 t v k=1 t v k=1
+J J Z q<k « 2k 'dsdt + J J £ q<k '«U <k 'dsdt + J J Z q<k '«u<k 'dsdt +
t s k=1 t s к=1 t s к=1
3 3 3
+J JZ f(k '«u(k ' dsdt + J JZ f<k '«u <k ' dsdt + J JZ f(k '«u3( k ' dsdt
s. * t s.
To determine the variation in the work of external and surface forces of a three-layer rod we performed analytical calculations: variation, differentiation, integration, and reduction of similar terms.
Variations of kinetic, potential energy and work of external and surface forces of a three-layer rod are substituted into the Ostrogradsky-Hamilton variational principle (1). As a result, we obtain the variational equation of a three-layer rod.
Conclusion
It is known that three-layer structural elements are increasingly used in modern industries, so there is a constant need to develop methods for their calculation in new operating conditions. This article defines variations in kinetic and potential energy, and variations in the work of external body and surface forces of a three-layer rod. The Ostrogradsky-Hamilton principle is applied to determine the variation of kinetic and potential energy and the work of external body and surface forces. The development of mathematical model and computational algorithm for the study of the stress-strain state of a three-layer rod will be given in the next articles.
t
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