The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity

Мірошніков, В Ю; Denysova, Tetiana; Protsenko, Volodymyr

doi:10.32347/2410-2547.2019.103.208-218

Cited by 5 publications

(6 citation statements)

References 3 publications

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“…This limit is taken into account in [30], where the problem is solved for a layer with one cylindrical cavity in displacements. A similar stress problem is considered in [31] and of a mixed type in [32]. A two-layer composite with a cylindrical stress cavity is considered in [33].…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…A two-layer composite with a cylindrical stress cavity is considered in [33]. However, the methods proposed in [28][29][30][31][32][33] do not allow one to take into account longitudinal inclusions.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Based on the conditions of statics, with only stresses specified at the layer boundaries, the equilibrium equations should be satisfied [31] ( )…”

Section: Main Research Hypothesesmentioning

confidence: 99%

“…On the limits of the inclusions with the layer, the conjugation conditions are set [31] ( ) ( ) 0 , , ,…”

Section: Main Research Hypothesesmentioning

confidence: 99%

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Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Miroshnikov

Savin

Younis

et al. 2022

EEJET

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An analytical-numerical solution of the spatial problem of elasticity theory for a composite in the form of a layer with two longitudinal endless continuous cylindrical inclusions is proposed. Homogeneous, isotropic materials of the layer and inclusions differ from each other in the modulus of elasticity and Poisson's ratio. Normal stresses are set on the upper and lower boundaries of the layer. The object of study is the stress state of such a composite. The problem is the lack of a high-precision method for calculating multiply connected bodies of this type. The solution of the problem is based on the generalized Fourier method for the Lame equations in various coordinate systems. The problem is reduced to an infinite system of linear algebraic equations, which is solved by the reduction method. In a numerical study, the stress state was obtained inside the composite bodies, within their conjugations, and on the isthmus between inclusions. It has been established that extreme stresses sρ=–0.9306 MPa, sf=–0.5595 MPa, tρf=–0.315 MPa occur on the mating face. Analysis of the stress state indicates the need to take into account the normal stresses on the mating surface. This is due to the presence of a binder, which may differ in physical characteristics from the main components of the composite. The results have logical physical correctness and, in simplified versions, are fully consistent with the results of similar problems from other approved sources. In the work, the transition formulas in the basic solutions between different coordinate systems, the conjugation conditions for different bodies, and the strict fulfillment of the equilibrium conditions for given boundary functions are simultaneously applied. This made it possible to obtain a high-precision solution of a new problem in the theory of elasticity for a layer with cylindrical inclusions and given only stresses on the boundary surfaces. The proposed method of calculation can be applied in the design of structures made of fibrous composites in the aircraft industry and construction.

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Miroshnikov

Savin

Younis

et al. 2022

EEJET

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“…Using this method, the following problems have been solved: for a layer with a spherical cavity [9], for a half-space with longitudinal cylindrical cavities [10 -11], for a cylinder with cylindrical cavities and inclusions [12 -16]. Problems for a layer with one or several cylindrical cavities are considered in [17 -20], for a layer with an elastic inclusion-in [21,22].…”

Section: Introductionmentioning

confidence: 99%

Investigation of the second main problem of elasticity for a layer with n cylindrical inclusions

Miroshnikov¹,

Денисова²

2021

OMTC

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When designing structures in the form of a reinforced layer, one has to deal with the situation when the reinforcement bars are located close to each other. In this case, their influence on each other increases. In order to obtain the stress-strain state in the contact zone of the layer and the inclusion, it is necessary to have a method that would allow obtaining a result with high accuracy. In this work, an analytical-numerical approach to solving the spatial problem of the theory of elasticity for a layer with a given number of longitudinal cylindrical inclusions and displacements given at the boundaries of the layer has been proposed. The solution of the problem has been obtained by the generalized Fourier method with respect to the system of Lame's equation in local cylindrical coordinates associated with inclusions and Cartesian coordinates associated with layer boundaries. Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method. As a result, displacements and stresses have been obtained at different points of the considered medium. When the order of the system of equations is 6, the accuracy of fulfilling the boundary conditions was 10-2 for values from 0 to 1. Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the reduction method, which is confirmed by the high accuracy of fulfilling the boundary conditions. In the numerical analysis, variants of the layer with 1 and 3 inclusions have been compared. The result has shown that close placement of reinforcement bars increases stresses on the surface of these inclusions. The values of stresses on the layer contact surfaces with inclusions have also been obtained. The proposed solution algorithm can be used in the design of structures, the computational scheme of which is the layer with longitudinal cylindrical inclusions and displacements specified at the layer boundaries.

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Analysis of the Stress State of a Layer with Two Cylindrical Elastic Inclusions and Mixed Boundary Conditions

Miroshnikov¹,

Savin²,

Hrebennikov³

et al. 2022

J. of Mech. Eng.

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The spatial problem of the elasticity theory is studied for a layer with two infinite circular solid cylindrical inclusions that are parallel to each other and to the layer boundaries. The physical characteristics of the layer and the inclusions are different from each other and they are homogeneous, isotropic materials. The spatial function of stresses is given at the upper boundary, and the function of displacements is given at the lower layer boundary. Circular cylindrical elastic inclusions are rigidly connected to the layer. It is necessary to determine the stress-strain state of the composite body. The problem solution is based on the generalized Fourier method, which uses special formulas for the transition between the basic solutions of the Lamé equation in different coordinate systems. Thus, the layer is considered in the Cartesian coordinate system, the inclusions – in the local cylindrical ones. Satisfying the boundary and conjugation conditions, systems of infinite integro-algebraic equations were obtained, which were subsequently reduced to linear algebraic ones. The resulting infinite system is solved by the reduction method. After deter-mining the unknowns, it is possible to find the stress values at any point of the elastic composite body. In numerical studies, a comparative analysis of the stress state in the layer and on the surfaces of inclusions at different distances between them is carried out. The analysis showed that when the inclusions approach each other, the stress state in the layer practically does not change. However, its significant change is observed in the bodies of inclusions, so with dense reinforcement ((R1 + R2) / L > 0.5), it is necessary to take into account the distances be-tween the reinforcing fibers. At stress values from 0 to 1 and the order of the system of equations m=10, the accuracy of meeting the boundary conditions was 10-4. With an increase in the system order, the accuracy of meeting the boundary conditions will increase. The given analytical-numerical solution can be used for high-precision determination of the stress-strain state of the given type of problems, and also as a reference for problems based on numerical methods.

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The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity

Cited by 5 publications

References 3 publications

Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Investigation of the second main problem of elasticity for a layer with n cylindrical inclusions

Analysis of the Stress State of a Layer with Two Cylindrical Elastic Inclusions and Mixed Boundary Conditions

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